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During its daily course above the horizon the Sun appears to describe a circular arc. Supplying in his mind's eye the missing portion of the daily circle, the Greek astronomer could imagine that his real eye was at the apex of a cone, the surface of which was defined by the Sun's rays at different times of the day and the base of which was defined by the Sun's apparent diurnal course. Our astronomer, using the pointer of a sundial, known as a gnomon, as his eye, would generate a second, shadow cone spreading downward. The intersection of this second cone with a horizontal surface, such as the face of a sundial, would give the trace of the Sun's image (or shadow) during the day as a plane section of a cone. (The possible intersections of a plane with a cone, known as the conic sections, are the circle, ellipse, point, straight line, parabola, and hyperbola.)
However, the doxographers ascribe the discovery of conic sections to a student of Eudoxus's, Menaechmus (mid-4th century bce), who used them to solve the problem of duplicating the cube. His restricted approach to conics—he worked with only right circular cones and made his sections at right angles to one of the straight lines composing their surfaces—was standard down to Archimedes' era. Euclid adopted Menaechmus's approach in his lost book on conics, and Archimedes followed suit. Doubtless, however, both knew that all the conics can be obtained from the same right cone by allowing the section at any angle.
The reason that Euclid's treatise on conics perished is that Apollonius of Perga (c. 262–c. 190 bce) did to it what Euclid had done to the geometry of Plato's time. Apollonius reproduced known results much more generally and discovered many new properties of the figures. He first proved that all conics are sections of any circular cone, right or oblique. Apollonius introduced the terms ellipse, hyperbola, and parabola for curves produced by intersecting a circular cone with a plane at an angle less than, greater than, and equal to, respectively, the opening angle of the cone.
Calculation
In an inspired use of their geometry, the Greeks did what no earlier people seems to have done: they geometrized the heavens by supposing that the Sun, Moon, and planets move around a stationary Earth on a rotating circle or set of circles, and they calculated the speed of rotation of these supposititious circles from observed motions. Thus they assigned to the Sun a circle eccentric to the Earth to account for the unequal lengths of the seasons.
Question: What was the purpose of the gnomon in the text? Answer: To generate a second, shadow cone that intersects with a horizontal surface to give the trace of the Sun's image during the day
Question: What is the reason given for the loss of Euclid's treatise on conics? Answer: Apollonius of Perga reproduced known results more generally and discovered many new properties of the figures | 677.169 | 1 |
This meets the definition of the tangent of the angle whose opposite side is -24 and whose
adjacent side is +7. Notice how this falls into quadrant IV = Since this ratio is the
tangent, by applying arctangent on a calculator, we find that the angle is -73.73979 degrees.
The polar form can be written in two ways:
25/-73.73979
which means a magnitude of 25 units with an angle found by rotating the magnitude
clockwise from positive portion of the x-axis by the amount 73.73979 degrees.
The angle can also be established by rotating (360-73.73979 or 286.26021 degrees)
counter-clockwise from the positive portion of the x-axis. This would be written as:
25/286.26021
I hope this helps you to understand the relationship between the Cartesian and polar representations
of points and specifically how to change from rectangular to polar form.
You may also find it helpful to make a rough sketch of the rectangular point (7,-24) and
draw the magnitude (radius) from the origin to this point. Measure out 7 on the positive
x-axis and the vertical distance from the point (7,0) to the point (7, -24). This will
help you to see the right triangle (0,0) to (7,0), from (7,0) to (7, -24), and then from
(7, -24) to (0,0). The two points (7, -24) to (0,0) are the ends of the radius and the
Magnitude of this line can also be found using the formula for the distance between two points.
the tangent of the angle is the ratio of the length of the line (7,0) to (7, -24) over
the length of the line from (0,0) to (7,0). Such a rough sketch will help you to visualize
the problem.
You can put this solution on YOUR website!
Multiply both the numerator and the denominator by
The denominator multiplies out to become 7 and the numerator becomes
So the answer is
The reason for simplifying the original problem, is that convention says to never leave a
radical in the denominator. Get rid of it by multiplying it by itself.
Hope this helps you.
You can put this solution on YOUR website! Recognize that by definition and
From this we can develop the sequence:
Note that this series is i, -1, -i, 1, i, -1, -i, 1, i, ....
We can use this table along with some rules of exponents to simplify the terms in the problem.
Let's try to simplify . We use one of the laws of exponents to rewrite
the this as . From the above table you can see that
Substitute this to get . We can further use a law of exponents
Question: What is the angle in degrees, as calculated using arctangent? Answer: -73.73979 degrees
Question: What is the tangent of the angle in the given problem? Answer: -24/7
Question: In which quadrant does the angle lie? Answer: Quadrant IV | 677.169 | 1 |
Question 290049: Solve for all possible triangles that satisfy a=9, b=13, measure of angle b=67 degrees. Round all values to the nearest tenth.
I know it is a lot of work, so i did part of it already :]
Since you know 2 sides and an angle. I can use law of sine to find a second angle. Since all the interior angles of a triangle is equal to 180 degrees, I can find the third angle. After this, I'm not too sure how to find the possible "multiple" types of triangles there are. Please help! Thank-you! Click here to see answer by dabanfield(803)
Question 290330: i'm doing SOH CAH TOA it uses sin, cos and tan my teacher trys to explain were everything goes but i just get confused. my question is how do you set it up? and when doing the equation how do you know when to put either the x over the number or the number over the x? for example: a ladder is leaning on a building the ladder is 18 ft from the building and the angle of the ladder from the ground is 22. Click here to see answer by stanbon(57984)
Question 290515: Hello,
We are given a triangle ABC. BC=1m,AC=6,6m and angle C=60degrees.
We draw a segment BD From B cutting AC on point D.The angle CBD=90degrees.
The triangle CBD is then rectangle on B. We are asked to find out the angle ABD.
I hope I am being clear so far.
So far, I've found out:
CBD is a triangle rectangle with C=60degrees then angle D=30degrees.
Also, BC=1m then CD=2m and BD=rt3m.
The angle ADB= 180-30=150 degrees and AD=AC-CD=6,6-2= 4,6cm.
So I know that the triangle ADB has angle =150degrees, BD=rt3m and AD=4,6m.
That is where I stop.
Thanks in advance for your help. Click here to see answer by richwmiller(9144)
Question 290649: If two triangles are similar how do i fond the value of the variable richwmiller(9144) Alan3354(31538)
Question 292 Click here to see answer by stanbon(57984)
Question 292 Click here to see answer by stanbon(57984)
Question: In the fourth question, what is the relationship between the two similar triangles? Answer: They are similar, meaning their corresponding angles are equal and their corresponding sides are in proportion. | 677.169 | 1 |
The Colour Field
Deconstructing ColourThu, 02 May 2013 14:26:14 +0000en-UShourly1 and Display
02 May 2013 14:26:14 +0000Melinda_1788 – physically, it is one of the strongest shapes in construction- think the pyramids.
Without going into a maths lesson, here are three most commonly used triangles that achieve a balanced composition, each attaining different types of symmetry:
Equilateral triangle, this achieves a symmetricalcomposition. It's a traditional method that we are innately comfortable with. Generally speaking it uses the same object repeated in the same positions at equal distances.
Scalene triangle, my default composition, giving an asymmetrical composition. This is when one side of the composition is higher than the other. Balance is achieved using some dissimilar objects that have equal visual weight. Visual weight refers to the physical space an item consumes. An asymmetrical composition is more casual and less contrived in feeling but it's more difficult to achieve. It suggests movement and leads to a more dynamic display.
Left or right angled triangles, also will give anasymmetrical composition. They are more location specific methods that tend to be placed with the vertical line against the wall or tall object. The highest point is generally positioned at the furthest point from the direction in which you approach.
Not a triangle, but worth a mention when discussing the various types of symmetry, is Radial symmetry. This is when all the objects are arranged around a center point. Not commonly used but can provide an interesting counterpoint if used appropriately. Definitely the method to use on a centrally placed entrance or hall table and dining tables.
What all triangles share (apart from three angles which I haven't gone into) is a high point and two low points and this is what any design needs, whether you are displaying on a shelf or trying to tidy a bookcase, even designing the interior of a room. Why? Because in visual terms, triangles create movement.
Displays can consist of your prized collection of spoons as in the book Bowerbird to old postcards and tools shown in The Natural Home. Think laterally about the item used to display collections on, for example, a vintage ironing board or an old hospital bed like in the book Creative Display. Take inspiration for books like Northern Delights and exhibit collections in areas of the home usually void of interest.
Rethink how you usually display and arrange everyday objects like flowers using books like Bringing Nature Home. These small changes create a big impact along with giving an insight into your personality.
Sharon Blair- Sydney
My favourite colour is Kournikova Yellow. It's Autumn in Sydney & during Easter we had a burst of Summer weather. Rain has started and Summer is over but I have a wall painted Kournikova Yellow to remind me.
]]> a Colour Field Holdall
Question: What type of composition does an equilateral triangle achieve? Answer: Symmetrical composition
Question: What does the author use as an example of a unique display item? Answer: A vintage ironing board or an old hospital bed | 677.169 | 1 |
But where the point that they do intersect – and I'm doing it really horrible – The point at which they do intersect would be equidistant between those two points, right? And another way of thinking about it, it would be another point that's equidistant between the two points, because when you do it first with P and you draw that circle, you're saying, "Oh, both of these points are going to be equidistant from P", right? Just by definition, right? This is a circle and that is a constant radius. And then if you would take each of those points and draw circles or draw arcs – So, let say from that one you would draw an arc like that. And from that one, you draw an arc like that. You'd say, "Wow, this point is also going to be of a constant distance from both of them." So, if I were to draw a line between both of these points using a straight edge, that line will be perpendicular to line L. So, if I were to just do that, that would be perpendicular. So, I think D is the first step.
All right. Problem 57. Which triangle can be constructed using the following steps? Okay, this is interesting. A lot of compass work here. Put the tip of the compass on point A. So, we go down there. So, they put the tip of the compass on point A. Open the compass so that the pencil tip is on point B. Okay, pencil tip is here. Draw arc above AB so then, they drew this arc right here. Draw arc above AB. That's this thing that I'm trying to color in. Fair enough. Without changing the opening, put the metal tip on point B. So now, you put the pivot there and draw an arc of intersecting – and draw an arc intersecting at point C. So now, they want us to draw that second – Let me do that another color. They're going to draw that second arc, right? Okay. Now, draw AC and BC. All right. Now, what have we drawn?
So when you draw this first arc, I mean an arc it's – we know we draw is like a semicircle, right? The radius is constant. So if the radius is constant, you know that this distance right here – that distance is going to be equal to this distance, right? They're just both radiuses of this semicircle or of this arc. They're just radiuses, right? They're equal to the length of the opening of our compass. So, that is going to be equal to that. And then when you put the pivot here, and you keep the distance the same – So now, the pencil edge goes here, right? The distance is still there and now when you do this arc, you now know that this length is equal to this length because now, they're both radiuses – They're both radiuses of this second arc. So now you know that three sides are equal. So, we have to be dealing this. This is an equilateral triangle. Equilateral – D. Okay.
Question: What are the final steps in constructing the triangle? Answer: Drawing lines AC and BC.
Question: What type of triangle is constructed using the given steps? Answer: An equilateral triangle.
Question: What is the distance between the two points on the arc? Answer: Equal to the length of the compass opening. | 677.169 | 1 |
in Triangle?
Hi!
Well, my problem is the following:
I have a point p(x|y) and three points a,b,c forming a triangle. How can I check, if the point p is in the triangle??
Maybe someone has a good, working function?
Please help!
TheBlob!
Re: Point in Triangle?
I have a simple method :
TEST the point P is in triangle ABC or not
add the area of PAB,PBC,PCA ,if the total is equal to the area of ABC ,then we know the point P is in the triangle ,otherwise is out.
right?
Question: What is the area of the triangle PAB in the assistant's method? Answer: The area of the triangle PAB is not given in the text. | 677.169 | 1 |
Six Frequency Figures
There are two Six Frequency Figures, the Great Rhombicuboctahedron and the Great Rhombicosidodecahedron. Each shape is made by cutting the line segments of the Two Frequency Figures into three parts in such a way that when all of the new points are connected all sides are of an equal length. Like the Truncated figures it is better to think of this as cutting off the corners.
In order to make the Great Rhombicuboctahedron a Cuboctahedron's line segments must be cut up in the way described above. Like in the Three Frequency Figures this method doesn't give the exact shape. The shape must squished a little to become a true Great Rhombicuboctahedron.
The Great Rhombicosidodecahedron comes from cutting up a Icosidodecahedron's line segments in a similar way. Once again the shape needs to be squished down to become a true Great Rhombicosidodecahedron.
Question: What is the main difference between the method of creating the Three Frequency Figures and the Six Frequency Figures? Answer: The method for the Six Frequency Figures doesn't give the exact shape, requiring some adjustment (squishing) to become a true figure | 677.169 | 1 |
Glide Reflections and Glide Reflective Symmetry
A last type of symmetry is glide reflective symmetry which results from the transformation calledglide reflection.
A glide reflection is actually a combination of a reflection and
a translation. Whether the reflection happens first or second
does not matter. The figure that results after a reflection and
translation is simply called the glide reflection of the original
figure. (Notice how "glide" refers to the translation part of
the combination.)
The same terms that apply to reflections and translations apply
to glide reflections. An axis is needed to perform the reflection
and a magnitude and direction are needed to perform the translation.
In this example of a glide reflection, the reflection is performed
first, and the translation performed second.
Here are two examples of glide reflections applied to some other
shapes. The original shape together with its reflection is said
to have glide reflective symmetry.
Examples of glide reflections and glide reflective symmetry
What does it mean for a tessellation to have glide reflective
symmetry? If we can perform a glide reflection to a tessellation
that such that the result is the same as the original tessellation,
then the tessellation has glide reflectional symmetry. An example
is as follows:
A tessellation that has glide reflective symmetry. To demonstrate
why, it is first reflected along the red dotted lines. Then, it
is translated until the newly made copy matches the originalexactly.
(One interesting note for further explorations: Every tessellation
that has reflective and translational symmetry also has glide
reflectional symmetry. Do you see why? After performing the appropriate
reflection, we can perform the appropriate translation. The overall
transformation demonstrates that the tessellation also has glide
reflectional symmetry. However, not all tessellations with glide
reflection symmetry have reflective symmetry and not all have
translational symmetry.)
Real examples of glide reflective symmetry:
After trying the hands-on activity, use the templates on the templates
page to see why every regular, semiregular, and demiregular tessellations
has glide reflectional symmetry. Use the technique that was explained in the hands-on
activity.
Question: What is the term used to describe a figure after it has undergone a glide reflection? Answer: The glide reflection of the original figure.
Question: Which comes first in the given example of a glide reflection: reflection or translation? Answer: Reflection.
Question: What does it mean for a tessellation to have glide reflective symmetry? Answer: If a glide reflection can be performed on the tessellation such that it matches the original, then it has glide reflective symmetry. | 677.169 | 1 |
MODERATORS
Hi, hoping for a quick answer, I'm trying to employ this technique and for the life of me I can't remember what it's called, and google is being an annoying little... well, I turn to you Reddit.
Basically, I have an area, which in this case is a circle but that's arbitrary. It's filled with points, or "stations" if you wanna call them that. I'm trying to divide up the tributary area so each point is "given" the optimal area between itself and the other points.
This is to my knowledge somewhat how divisions of an insects eye/wing are developed, or the location of rain gauges in a watershed, etc..
I'm just looking for the name of the process if division, I figured out/remember the process, but I can't remember what the principle is called.
Question: What is the user unable to remember? Answer: The name of the process or principle for dividing the area | 677.169 | 1 |
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real
Question: What is the relationship between the values of trigonometric functions for π–x and π+x? Answer: The values of trigonometric functions for π–x and π+x are co-function identities, meaning sin(π–x) = cos(x), cos(π–x) = sin(x), tan(π–x) = -tan(x), and similarly for π+x. | 677.169 | 1 |
This (hands-on) demo illustrates how a carpenter can draw an ellipse on wood or a sheet of wall board using simple tools. A jig can be used to demonstrate the technique and there are software animations to illustrate the use of the jig. — "A Carpenter Draws an Ellipse", mathdemos.gcsu.edu
An ellipse (shaded green) is a type of conic section. In mathematics, an ellipse (from the Greek word ἔλλειψις, which literally means "absence") is a closed curve on a plane, such that the sum of the distances from any point on the curve to two fixed points is a constant. — "Ellipse - New World Encyclopedia",
In geometry, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the cone's axis. — "Ellipse - Wikipedia, the free encyclopedia",
That page also contains some background information on conic sections and other topics that also applies to ellipses, that won't be repeated here. The ellipse is a typical oval, but a very particular one with a shape that is regular and can be exactly specified. — "Ellipse", mysite.du.edu
Ellipse is a family of curves of one parameter. Together with hyperbola and parabola, they make up the conic sections. Ellipse is also a special case of hypotrochoid. The eccentricity of a ellipse, denoted e, is defined as e := c/a, where c is half the. — "Ellipse",
It works from n=1 to n=100 for variable a b without noticeable gaps or overlaps for n>7 There is some slight overlapping at n=7 and a b=1 5 But I can fudge that by changing a b to 1 6 The approach is
Ellipse jpg
Geometry Ellipse Layout aka String Method Geometry Finding the Slope at any Point on Ellipse Formula for Slope at any Point on an Ellipse
ELLIPSE png
The figure below represents a planet s prograde orbit around the sun dot in center a is called the semimajor axis of the planet s orbit It is also the average distance from the planet to the sun e is the eccentricity of the orbit At closest approach to the sun
Ellipse To use any of the clipart images above including the thumbnail image in the top left corner just click and drag the picture to your desktop You may also control click Mac or right click
digital CTG IWAKI KERRY TANAKA XY plotter IZUHARA
ELLIPSE png
ELLIPSE png
ellipse gif
旋涡星系
Ellipse To use any of the clipart images above including the thumbnail image in the top left corner just click and drag the picture to your desktop You may also control click Mac or right click
Question: What is the primary tool used to draw an ellipse on wood or a sheet of wall board in the given demo? Answer: A jig | 677.169 | 1 |
0 00 From this table we can plot a smooth graph such as the one shown below Like the circle it is easy to generalize Equation 9 1 7 to the case of ellipses not centered at the origin but still with horizontal major axis and vertical minor
Videos
Ellipses (Part I) Basic introduction to Ellipses including general formula.
Imogen Heap - Ellipse Preview Which song is your favorite? If you haven't pre-ordered, don't forget to do so. Preview in mp3 on
GIMP - Rectangle and Ellipse Select Tool. Things you didn't know! (Photoshop marquee tool) In this GIMP how to video I'll be showing you all the interesting tool options of the GIMP with the rectangle select tool and ellipse select tool. I'll be showing you how to make shapes in GIMP and how to make selections. How to feather selections and how to use the modes, like add and subtract. I made this video for beginners but maybe more advanced user may pick up something too...
CARSON WOODWORKS - Make an Ellipse Mirror Geoffrey Carson of Carson Woodworks makes an Ellipse shaped wall mirror from Birdseye Maple. Some procedures are left to the imagination and ingenuity of the viewer. You can email me if you have any questions. geoff@
Lecture - 23 Matrix Times Circle : EllipseEquation Of the Ellipse Check us out at An ellipse is the set of all points in a plane such that the sum of the distances from T to two fixed points F1 and F2 is a given constant, K. TF1 + TF2 = K F1 and F2 are both foci(plural of focus) of the ellipse. The major axis is the segment that contains both foci and has its endpoints on the ellipse. These endpoints are called the vertices. The midpoint of major axis is the center of the ellipse. The minor axis is perpendicular to the major axis at the center, and the endpoints of the minor axis are called co-vertices. The vertices are at the intersection of the major axis and the ellipse. The co-vertices are at the intersection of the minor axis and the ellipse.
Graphing Ellipses The video shows the standard form of the equation of an ellipse. It shows how the h value shifts the ellipse left and right while the k value shifts if up and down. It demonstrates graphing ellipses from a standard form equation. It explains the major axis and minor axis. It looks at vertical and horizontal ellipses. It does four standardized type test problems with ellipses. It shows the conics application of the TI 84 series calculators.
Drawing Ellipses and Cylinders in Perspective This tutorial demonstrates how to draw ellipses accurately in perspective. From there you can draw cylinders in different orientations in the proper perspective.
Draw A Perfect Ellipse Easy way to draw a perfect Ellipse. Go to mezon.biz for more!
Question: Which of the following is NOT a part of an ellipse? A) Foci B) Vertices C) Edges D) Co-vertices Answer: C) Edges
Question: What is the constant sum of distances from any point on an ellipse to its two fixed points called? Answer: K | 677.169 | 1 |
A quadrilateral, sometimes also known as a tetragon or quadrangle ( If the points do not lie in a plane, the quadrilateral is called a skew quadrilateral. — "Quadrilateral -- from Wolfram MathWorld",
A quadrilateral derives directly from a Quadrangle by the regrouping of the tops in two pairs. For each pair, the two tops are known as opposite and the segment which joined them (with dimensions of the quadrangle), is not regarded any more a side, but as a diagonal of the quadrilateral. — "Quadrilateral - SpeedyLook Encyclopedia",
Definition of quadrilateral in the Dictionary. Meaning of quadrilateral. What does quadrilateral mean? Proper usage of the word quadrilateral. Information about quadrilateral in the dictionary, synonyms and. — "What does quadrilateral mean? definition and meaning (Free",
Definition of quadrilateral in the Online Dictionary. Meaning of quadrilateral. Pronunciation of quadrilateral. Translations of quadrilateral. quadrilateral synonyms, quadrilateral antonyms. Information about quadrilateral in the free online. — "quadrilateral - definition of quadrilateral by the Free",
In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. The interior angles of a simple quadrilateral add up to 360 degrees of arc. — "Quadrilateral - Wikipedia, the free encyclopedia",
Quadrilateral - Meaning and definition quadrilateral. 1. A plane figure having four sides, and consequently four angles; a quadrangular figure; any figure formed by four lines. — "Quadrilateral - Encyclopedia",
A quadrilateral is a polygon that has exactly four sides. Some examples of quadrilaterals: Discussions of 2-D shapes sometimes refer only to the boundary (the line segments that form the edges of the figure) or to the interior as well. — "Quadrilateral - Thinkmath",
Images related images for quadrilateral
X
X
to the other end of it and the magenta rod to the other ends of the magenta and grey rods the free ends of the magenta green and grey rods must lie on the magenta green and grey circles Here the green is opposite the brown But of course it does not have to be like this it could be like this
inradii of the triangles ABD ABC BCD and ACD respectively Prove that EFGH is a rectangle and ra + rc = rb + rd See the proof
長方形 一般四角形 正三角形
X
Quadrilateral With Diagonals and Midpoints Joined To use any of the clipart images above including the thumbnail image in the top left corner just click and drag the picture to your desktop You may also control click Mac or right click
Quadrilateral Polygon To use any of the clipart images above including the thumbnail image in the top left corner just click and drag the picture to your desktop You may also control click Mac or right click
Question: Which of the following is NOT a synonym for a quadrilateral? A) Tetragon, B) Quadrangle, C) Pentagon, D) Quadrilateral Answer: C) Pentagon | 677.169 | 1 |
Quadrilateral Rap A rap to teach you about quadrilaterals. QUADRILATERAL RAP LYRICS Quadrilateral is a four sided polygon Let's draw some up so pick up yo crayons We got one side, two sides, three sides, four Let's start out with a square and then well talk more More, more Talk more More, more All squares have four All the angles are congruent so there known as equal angular You might even hear a square referred to as rectangular Now we got that covered we can move on to the kite Theres so much to say bout this shape we could be here till the night To the night, night To the night To the night, night Yo my name is Black Knight A kite has two pairs of adjacent and congruent sides If anybody disagrees you tell them they just lied Every convex kite seems to have an inscribed circle And if you dont believe me, man you crazy like Steve Urkle Urkle, Urkle Steve Urkle Urkle, Urkle Inscribed circle A trapezoid is the next shape on our list If you don't listen yo your gunna get dissed They got one pair of opposite parallel sides The other two are legs which will always collide Lide, lide Collide Lide, lide Got rules to abide Diagonals in a parallelogram bisect each other If you dont believe me go ahead and ask yo mother Pairs of opposite angles are also congruent If you listen to these properties youll know it fluent Fluent, fluent Fluent, fluent Opposite angles are also congruent Yo look at the time; Ive reached the end of my rap Thank you, thank you no need to clap See you all later hope you have ...
Area of Special quadrilaterals (Simplifying Math) In this mini-lesson I demonstrate how to find the area of parallelograms and trapezoids.
Quadrilateral optical recognition A small application developed in C++ and OpenCV detecting quadrilateral shapes in real-time.
QUADRILATERAL SONG This song is System of a Down;s "Chop Suey", but it's about quadrilaterals.
Quadrilaterals & Polygons Math
Area of triangles and quadrilaterals How to find the area of triangles, rectangles, parallelograms, rhombuses and trapeziums.
What Guarantees That a Quadrilateral Is a Parallelogram? : Measurements & Other Math Calculations Subscribe Now: Watch More: For a quadrilateral to be considered a parallelogram, a few key things need to be true. Learn what guarantees that a quadrilateral is a parallelogram with help from a physics professional in this free video clip. Expert: Julia Lundy Filmmaker: Julia Lundy Series Description: Mathematics is a large and varied topic with many different facets, so it can only be natural to feel a bit overwhelmed from time to time. Get tips on performing and solving a variety of different math problems and functions with help from a physics professional in this free video series.
Quadrilateral Rap rap by bob
Question: Which song is mentioned to be about quadrilaterals? Answer: System of a Down's "Chop Suey"
Question: Which shape has an inscribed circle? Answer: Kite
Question: What is the main property of a trapezoid's sides? Answer: It has one pair of opposite parallel sides | 677.169 | 1 |
Flag this because it ...
In triangle PQR, sides PQ and QR have lengths 5 and 11, respectively. Triangle PQR has the same perimieter as equilateral triangle STU. If the lengths of side ST is an integer, which of the following could be the length of side PR?
Answers
You have not listed the possible lengths, but we can eliminate some integers: 1 through 5.
Let x be the length of side PR
16 + x is the perimeter of triangle PQR
Triangle STU is equilateral so let its sides be Y and Y must be an integer.
16 + x = 3Y
x = 3Y -16
Y must therefore be 6 or larger because a negative length does not make sense.
Question: What is the perimeter of triangle PQR? Answer: 16 + x, where x is the length of side PR. | 677.169 | 1 |
Therefore ED equals GD. And DF is common, therefore the two sides ED and DF equal the two sides GD and DF, and the angle EDF equals the angle GDF, therefore the base EF equals the base GF, the triangle DEF equals the triangle DGF, and the remaining angles equal the remaining angles, namely those opposite the equal sides.
Therefore, if two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides.
Q.E.D.
This is a side-angle-side similarity theorem analogus to side-angle-side congruence theorem I.4.
Here's a summary of the proof. Construct a triangle DGF equiangular with triangle ABC. Then triangle DGF is similar to triangle ABC (
VI.4), and that gives us the proportion
BA:AC = GD:DF.
But we have assumed the proportion
BA:AC = ED:DF,
and these two proportions together give us
GD:DF = ED:DF
(V.11), from which it follows that GD = ED (V.9). Therefore triangles DEF and DGF are congruent, and the rest follows easily.
Question: Which theorem is used to prove that triangle DGF is similar to triangle ABC? Answer: VI.4 | 677.169 | 1 |
Mathematical Proofs: The Cosine Rule
How many times have your teacher given you a formula to use in class? How many times have you wondered where it came from? Here I will show you one of the most fantastic formula used in Mathematics! The Cosine Rule. A step by step guide as to how the Cosine rule was derived. Feel free to rewind as well as the video and learn how to prove the Cosine rule. Enjoy!!
only half of a success. because drawing a unitcircle over the angle A would have proven useful.
and also exploring this case with what if we draw the line h from angle C, wouldve been interesting
but thanks anyway
Question: Which of the following is NOT a part of the text? (a) A formula (b) A step-by-step guide (c) A joke (d) A promise Answer: (c) A joke | 677.169 | 1 |
Solid Angle
Solid Angle
Hi, could someone explain to me the concept and calculation of Solid Angle? I don't think we've actually covered it in our Vector Calculus lectures and I have a question to do on it!!! Tried searching on the web, but not much information and I really don't understand it.
Also, my question is:
"Calculate the Solid Angle of a cone of half-angle 'alpha'".
What is the half-angle in a cone?
A transversal section through a cone,if done as to contain the axis reveals a triangle.My gues is that in your case,the triangle is isosceles...The 2 rightangle triangle (congruent) each has an angle [itex]\alpha [/itex]...So the total angle is [itex] 2\alpha[/itex]...Use the definition of the solid angle and compute it.
Question: What is the concept of a solid angle? Answer: A solid angle is a measure of how much of the surface of a sphere it takes to completely surround a point in three-dimensional space. | 677.169 | 1 |
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About this topic
Triangles have fascinated people since ancient times. We see the triangle shape being used in many buildings and constructions in the world around us. In this series of lessons, we study the properties of triangles and find ways of identifying triangles by their properties. We construct, measure and compare the lengths of the sides and the sizes of the angles. Learners discover important properties of seven types of triangles and investigate their lines of symmetry. We also investigate two triangle theorems and use similarity of triangles to solve problems.
Question: What shape is the focus of this series of lessons? Answer: Triangles | 677.169 | 1 |
The ratios between sides of a triangle: sine {sin} (the side opposite the angle over the hypotenuse), cosine {cos} (the side adjacent to the angle over the hypotenuse), and tangent {tan} (opposite over adjacent); and their inverses: secant {sec} (hypotenuse over opposite), cosecant {cosec} (hypotenuse over adjacent), and cotangent {cot} (adjacent over opposite). (See Construction and Use of a Sundial and Observations Using a Gnomon)
Originally, the names given to the 6 possible ratios between pairs of sides in a right-angled triangle (sine, cosine, tangent, cotangent, secant, cosecant). Usually the triangle is drawn with resting on one of its shorter sides, and these functions are viewed as depending on the bottom acute angle (angle smaller than 90 degrees). Later the definitions were extended for any angle, using the unit circle. Though initially introduced as a tool of land-surveying, today trigonometric functions play a key role in many areas of mathematics.
Question: What is the hypotenuse in the unit circle used for? Answer: Extending the definitions of trigonometric functions for any angle | 677.169 | 1 |
Examples triangle's examples
Our Triangle team will keep you updated on events, new construction, These upscale apartment homes are part of an impressive 22-acre mixed-use community, also named, The Triangle, that will unfold to include 529 apartment homes; over 120,000 square feet of retail, commercial, restaurant. — "Triangle Austin Apartments | Austin Lofts | Austin Townhomes",
Triangle TRACKS links to 2,000+ sources across the region focused on the improvement of children's lives. — "Triangle TRACKS",
Setting aside the economic arguments against the big box retailer for one day let's have a brainstorm on how the new Walmart could fit within the urban context of the neighborhood. [caption id. — "Walmart brainstorm | The Triangle",
Triangle United's U16 Green Girls won their age division's top bracket in the Lake Norman Fall Classic soccer tournament this past weekend. — "Triangle teams shine in tournament play - Club Soccer",
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two).[2] An isosceles triangle also has two. — "Triangle",
Triangle is truly your single-source for packaging automation. This unique advantage makes Triangle the only company in the world that manufactures vffs baggers, combination weighers and bag-in-box systems, all under one roof. — "Food Packaging | VFFS, Weighing, Bag-In-Box",
Every triangle has three sides and three angles, some of which may be the same. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior. — "Triangle -- from Wolfram MathWorld",
In an equilateral triangle all sides are equally long. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°. In an isosceles triangle two sides are equally long. An isosceles triangle also has two equal internal angles. — "Triangle - Definition",
for geometrical aspects of triangles, see Category:Triangle geometry for musical aspects of triangles, see Category:Triangle (instrument) To display all parents click on the ". — "Category:Triangles - Wikimedia Commons",
Check out these party pics and see where you may want to hang this weekend in the Triangle! Holiday Events. See our list of upcoming holiday events. Greg Cox's Hot List: Java Joints. Looking for a jolt to get you through those cold, dark mornings?. — "Events, movies, restaurants, contests, community - Raleigh",
The Triangle United Junior YDA Program is new program for the 2010-2011 season. 91 Triangle United Gold Boys and Their Legacy. Over the past 4 years, the 91 Triangle United Gold Boys team has helped put Triangle United on the map by their outstanding play on the field and their love for. — "Triangle United Soccer",
Question: What is the measure of each internal angle in an equilateral triangle? Answer: 60 degrees
Question: Which soccer team won their age division's top bracket in the Lake Norman Fall Classic tournament? Answer: Triangle United's U16 Green Girls
Question: What is the total square footage of retail and commercial space planned for the Triangle community? Answer: Over 120,000 square feet
Question: How many apartment homes are planned for the Triangle community? Answer: 529 apartment homes | 677.169 | 1 |
11/27 Triangle Singles Club Dance in Raleigh at Raleigh, North Carolina, United States. Come join us for a fun night of dancing and mingling with other. — "11/27 Triangle Singles Club Dance Raleigh - 11/27 Triangle",
Pascal's triangle ( ) n. A triangle of numbers in which a row represents the coefficients of the binomial series. — "Pascal's triangle: Definition from ",
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas. — "Triangle - Wikipedia, the free encyclopedia",
Provides up-to-date, in-depth research on the mystery of the Bermuda Triangle and the disappearance of Flight 19. — "Bermuda ", bermuda-
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon. — "Triangle", schools-
Assists people with disabilities in maximizing their sense of dignity and self worth by providing quality vocational evaluation and training, employment, social development, and residential services. — "Triangle, Inc", triangle-
Triangle Fraternity Headquarters will be closed beginning at 12:00 noon EST on Wednesday, November 24 and all day Thursday and Friday, November One full tuition scholarship will be awared to a Triangle for the 2011 voyage. Be sure to click READ MORE beneath the graphic of the ship for all of. — "Triangle Fraternity",
Adoption of a New Project Comparative Research on Major Regional Powers in Eurasia A new project Comparative Research on Major Regional Powers in Eurasia was adopted by the Ministry of Education and Science as a five year grant in aid for scientific
triangle jpg
I finally can feel a triangle and its purpose and while mine isn t even mediocre yet it s not as mysterious as it used to be That was my big revelation for the night
the six floors we had plenty of time to take new photos of all sorts of land marks The Getalife building and Chrysler building are just some of the attractions that I managed to photo
triangle jpg
are a number of basic variables to project delivery scope cost and time Not all these variable can be fixed and generally quality is in the middle as another non negotiable variable This can be related to a typical Agile Burndown in that the Horizontal Axis is Time the Vertical Axis is Scope and the burndown slope is a function of cost and time Keith Braithwaite and
Equilateral triangle To use any of the clipart images above including the thumbnail image in the top left corner just click and drag the picture to your desktop You may also control click Mac or right click
Question: What is the date and location of the Triangle Singles Club Dance? Answer: The dance is on 11/27 in Raleigh, North Carolina, United States.
Question: When will Triangle Fraternity Headquarters be closed? Answer: It will be closed from 12:00 noon EST on Wednesday, November 24 to all day Friday, November 26. | 677.169 | 1 |
DBSK Triangle Live Live Performance of the Korean band DBSK Ft. BOA and THE TRAX
The mystery of the Bermuda triangle The disappearance of US Navy flight 19 is still the most astounding mystery associated with the Bermuda triangle. Could it be solved by using modern means? Taken from TV show: "The Bermuda Triangle" available on DVD at Download legally in full at:
Triangle sun - Beautiful Official Music Video You can find out more about the band and its music at /trianglesun Director: Roman Jirnih DOP: Marat Adelshin Edit: Ivan Gaev Color correction: Salamandra studio Production: Guerilla shots
Proving Triangles are Congruent - - Math Help For a complete lesson on proving triangles are congruent, go to - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn the following postulates related to congruent triangles and triangle proofs. If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent (Side-Side-Side or SSS). If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent (Side-Angle-Side or SAS). If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent (Angle-Side-Angle or ASA). Students are then asked to determine whether given triangles are congruent, and name the postulate that is used.
The High Kings Auld Triangle The High Kings sing the irish folk classic The Auld Triangle.
New Order - Bizzare Love Triangle '94 This is the 1994 version of "Bizarre Love Triangle" by New Order which appears on the "Best Of" album from the same year.
Mystery of the Bermuda Triangle Science fiction or science fact? What are some of the possible causes of the mysterious disappearances in this area of the Atlantic Ocean?
Similar triangles Introduction to similar triangles
Seo taiji - Bermuda [Triangle] He released a new single 'Bermuda'. He's valuated as one of the krean legendary musicians. He's famous for his creative music, not limited by genre and good arranged marketing.. Also his baby face(Guess his age..if you know that..what a surprise...) and the perfact harmonious voice for his colorful music styles. In my opinion, he is without a match in his own unique music idea & tryout. This song is connected to the theme of ruined ***. I bet You will be addicted this song..^^
Weezer - Pink Triangle From the Pinkerton album
Sneak Peak of James Blunt on Sesame Street Enjoy this preview of James Blunt singing a parody of his hit song "You're Beautiful" titled "My Triangle" airing on August 31st, 2007 as part of the new season of Sesame Street!
Question: Who is the director of the music video "Triangle sun"? Answer: Roman Jirnih
Question: In which year was the 1994 version of "Bizarre Love Triangle" by New Order released? Answer: 1994 | 677.169 | 1 |
TETRAHEDRON (Gr. TErpa-, four, Spa, face or base), in geometry, a solid bounded by four triangular faces. It consequently has four vertices and six edges. If the faces be all equal equilateral triangles the solid is termed the "regular" tetrahedron. This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system. The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids)., and the trigonal pyramid of the hexagonal system, are examples of non-regular tetrahedra (see Crystallography). "Tetrahedral co-ordinates" are a system of quadriplanar co-ordinates, the fundamental planes being the faces of a tetrahedron, and the co-ordinates the perpendicular distances of the point from the faces, a positive sign being given if the point be between the face and the opposite vertex, and a negative sign if not. If (u, v, w, t) be the co-ordinates of any point, then the relation u+v-Fw-fit=R, where R is a constant, invariably holds. This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.
Related to the tetrahedron are two spheres which have received much attention. The "twelve-point sphere," discovered by P. M. E. Prouhet (1817-1867) in 1863, is somewhat analogous to the nine-point circle of a triangle. If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the mid-points of the segments of the perpendiculars between the vertices and their common point of intersection. This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
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Question: What are tetrahedral co-ordinates? Answer: Tetrahedral co-ordinates are a system of quadriplanar co-ordinates where the fundamental planes are the faces of a tetrahedron, and the co-ordinates are the perpendicular distances of the point from the faces.
Question: Which system does a regular tetrahedron belong to in crystallography? Answer: The cubic system. | 677.169 | 1 |
[Thinking "half of a side" in terms of perpendicular bisectors leads us to perpendicular bisectors of chords of the circumcircle. Thinking "half of an angle" suggests that angle bisectors could be used to identify the incircle and to locate the incenter. From half of a side or half of an angle, we get important ideas that underlie how chords and tangents are related to the constructions.]
Questions for StudentsAssessment Options
Have students construct the circumcenter and the incenter of a triangle. Provide students with a triangle that is not an isosceles triangle and not oriented in the same way as the triangle from the hospital problem.
As a journal entry, students should write a response to the following questions: How do the circumcenter and incenter involve bisecting the triangle's sides and angles? Why do these bisection processes makes sense for these constructions?
Extensions
Is it possible for the circumcenter and the incenter of a triangle to be the same point? Explain. [The circumcenter and the incenter would be the same point when all of the perpendicular bisectors of the sides and the angle bisectors of the triangle coincide. This happens when the triangle is an equilateral triangle. So, yes, it is possible.]
Perpendicular bisectors of the sides led to the circumcenter. The bisecting of the angles led to the incenter. What happens if we simply construct a line through the midpoint of each side and the opposite vertex? [The three lines intersect in a unique point. Subsequent discussion may lead to the description of the lines as the medians and the intersection point as the centroid. The sense of "half" comes into the discussion because each median divides the area of the original triangle in half, as can be seen in the Half Angle applet. (Because M is the midpoint of the base, it follows that the area of triangles AMB and CMB will always be equal. Both have height BM, and the bases of the triangles are equal (AM = CM), so the areas are equal: ½(AM)(MB) = ½(CM)(MB).]
Teacher Reflection
Where did students struggle in executing the constructions?
How did students use their ideas about "half" to explain why the constructions for circumcenters and incenters make sense
Question: How do the circumcenter and incenter involve bisecting the triangle's sides and angles? Answer: The circumcenter is found by constructing perpendicular bisectors of the sides, while the incenter is found by bisecting the angles.
Question: In which type of triangle do the circumcenter and incenter coincide? Answer: An equilateral triangle. | 677.169 | 1 |
I'm trying to write a procedure or function to find the direction in which a 2D triangle points. The triangle is assumed to be isosceles. While I can see the basic outline of what I want to do, making this into a procedure/function is proving harder. This is very basic Mathematica programming, I think... :(
Suppose that points p, q, and r are three arbitrary 2D points assumed to form an isosceles triangle:
Note that the direction is given as a vector (which carries the additional information about the distance from the triangle's center towards this extreme vertex). For other forms of output, this is readily converted into a unit vector (by normalizing it) or an angle.
Edit
Notice that for squat, flat isosceles triangles (where the apex angle exceeds 60 degrees), this solution determines that the "pointing direction" is from the center towards one of the base vertices, not towards the apex. If that is not the intention, then additional analysis is needed. One way is to determine that among all the possible directions from the center, the one most different from the other two is the one to pick:
In this implementation (which is written for clarity rather than brevity), s lists the three possible directions, l lists their lengths, and for each entry in l, a lists the differences in lengths between the other two. The last line returns the direction where that difference is the smallest (as a vector from the center to the vertex, exactly as before).
@R.M. You are correct--thank you--but not for the reason you might think! The problem is that Ordering is different for exact values than it is for numerical values. (Although this is well documented behavior, it's a subtle and nasty one IMHO.) One solution is to convert the norms to machine numbers. – whuberApr 16 '12 at 18:39
@whuber I wasn't referring to Ordering's behaviour... rather that the most distant vertex (or vertices) in such cases would be the two vertices whose subtended angles are equal, rather than the third vertex which is what the OP wants. – rm -rf♦Apr 16 '12 at 18:44
If the triangle is isosceles, the corresponding median line is orthogonal to the side. Therefore I think the following should be the most robust solution (note that it also works for triangles where the arrow tip angle is larger than 60 degrees, however it assumes the triangle is not degenerate; also for equilateral triangles it will return just one of the three possible directions):
The function takes a triangle as first argument, and compares its length with some precision (for your input data, 0.1% is a good choice of precision, so 0.001). Then calculates the direction as a vector (v) and return its argument with ArcTan's two-argument variant.
Mathematica is a registered trademark of Wolfram Research, Inc. While the mark is used herein with the limited permission of Wolfram Research, Stack Exchange and this site disclaim all affiliation therewith.
Question: What is the issue with determining the "pointing direction" for squat, flat isosceles triangles? Answer: For such triangles, the "pointing direction" is determined to be from the center towards one of the base vertices, not towards the apex. | 677.169 | 1 |
...that a triangle has 180 degrees is that back then when them things were assigned, a year only had 360 days; Xmas came 5 1/4 days earlier.
Methinks we morons here should team up and march on the "Mathematics High Place" demanding that this be changed to 400; much easier to handle 100 degrees quadrants (plus create employment making up sin/cosine tables).
Our triangles would now have 200 degrees; WOWEE! Imagine that: we will be able to have two 90 degree angles, so call on Big Pete Pythagoras twice to solve "c"; triangles will have new "standard" sides of a,c,c !!
Question: What is the advantage of having a 200-degree triangle? Answer: It would allow for two 90-degree angles | 677.169 | 1 |
Numberplay: Heights of a Triangle
Welcome to Numberplay — the week-long game of debate and discovery that also happens to be about math.
Here's how it works. Each week we take a simple math puzzle, solve it, and then explore the standard solution. Did the solution really work? Are there other solutions that would be just as good — or even better?
Reader Caleb put it well:
At first I thought the question was too easy, but the real beauty is in the discussion that arises so quickly after the answer.
And the discussion has just gotten even more beautiful — or at least more organized. Check out Gary Hewitt's latest Numberplay Commenting Fix.
Back to the puzzle. Our discussion this week starts off with the following:
The solution: reader deo got straight to the beauty of the problem. If Numberplay were scored I think the judges would have stopped right there and given deo all possible points.
Here's deo:
Thinking "out loud", on a plane, no, on a sphere yes.
That is, a side of three on a diameter can be connected by sides of one and two at a point above (or below) the diameter.
I think.
Deo was onto something with the sphere, which Rich in Atlanta and Hans, among others, explored in detail for most of the week. Adam proposed one of the early sphere-based solutions:
One configuration that could work (but I don't think could be called a triangle) would be the 3-side lying on the surface of a sphere and the 1- and 2-sides cutting across the interior. The result would look sort of like a lopsided ice-cream cone.
But wait. Let's return to deo's plane for a moment. You can't get out a piece of paper and and draw a triangle of sides 1, 2 and 3? Here's Giovanni Ciriani:
The answer is no, because the length of the longest side needs to be shorter than the sum of the other two.
Ah. What a great way to put it. "Bad triangle, good drawbridge," observed Joe Mahoney. Larry Eisenberg captured the moment with this limerick:
With the law of cosines, I see,
The angle opposite side 3
Is one eighty degrees
The question's a tease,
A triangle like that can't be.
But what about the flip side of the answer? How could a triangle have sides 1, 2 and 3? Among the many ideas that emerged:
Betsy: "Yes, if the triangle is the musical triangle instrument. One side must be open on a musical triangle by definition."
And could anything be more beautiful than this? Here's Dr W via e-mail:
Suppose we position three straight line segments of length 1, 2 and 3 in the (x,y,z) Cartesian coordinate system so all ends lie at points on each of the three coordinate axes, as follows:
Length 1 from A (0, 0, 1) to B (0, 0,0)
Question: Is the Numberplay game focused on solving mathematical puzzles? Answer: Yes.
Question: What is the primary goal of the Numberplay game, according to the text? Answer: To explore the standard solution of a simple math puzzle and discuss whether it's the best or only solution. | 677.169 | 1 |
Products of vectors.
The multiplication of vectors leads to two types of products, the dot product and the cross product.
The dot or scalar product of two vectors a and b, written a·b, is a real number |a||b| cos (a,b), where (a,b) denotes the angle between the directions of a and b. Geometrically,
If a and b are at right angles then a·b = 0, and if neither a nor b is a zero vector then the vanishing of the dot product shows the vectors to be perpendicular. If a = b then cos (a,b) = 1, and a·a = |a|2 gives the square of the length of a.
The associative, commutative, and distributive laws of elementary algebra are valid for the dot multiplication of vectors.
The cross or vector product of two vectors a and b, written a × b, is the vector
where n is a vector of unit length perpendicular to the plane of a and b and so directed that a right-handed screw rotated from a toward b will advance in the direction of n (seeFigure 2). If a and b are parallel, a × b = 0. The magnitude of a × b can be represented by the area of the parallelogram having a and b as adjacent sides. Also, since rotation from b to a is opposite to that from a to b,
This shows that the cross product is not commutative, but the associative law (sa) × b = s(a × b) and the distributive law
Question: What is the magnitude of the cross product of two vectors a and b represented by? Answer: The area of the parallelogram having a and b as adjacent sides.
Question: If a right-handed screw is rotated from vector a toward vector b, in which direction will it advance? Answer: In the direction of n. | 677.169 | 1 |
of the triangle with the base, and apply the Pythagorean Theorem. This applet
shows in a dynamic way how the theorem works.
Hold any vertex and move it around to see how the values in the
equation are updated, and how they comply with the theorem.
There are problems in which to find the area of the triangle, you can't use the Pythagorean Theorem,
or special right triangles; you are left with trigonometry as your only option. You may drag the
vertex in the triangle below to see how the trigonometric ratios are updated, and their
relationship with their inverses.
Part of solving the areas for special triangles consist in understanding basic definitions
for special segments in triangles. These are altitudes, angle bisectors, medians, and perpendicular
bisectors.
Practice in this applet dragging any of the vertices in the triangles, and read the definitions;
Question: What can you observe by dragging the vertex in the second triangle mentioned in the text? Answer: The update of trigonometric ratios and their relationship with their inverses | 677.169 | 1 |
Current location in this text. Enter a Perseus citation to go to another section or work. Full search
options are on the right side and top of the page.
PROPOSITION 2.
If the square on a straight line be five times the square on a segment of it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line.
For let the square on the straight line AB be five times the square on the segment AC of it, and let CD be double of AC; I say that, when CD is cut in extreme and mean ratio, the greater segment is CB.
Let the squares AF, CG be described on AB, CD respectively, let the figure in AF be drawn, and let BE be drawn through.
Now, since the square on BA is five times the square on AC, AF is five times AH.
Therefore the gnomon MNO is quadruple of AH.
And, since DC is double of CA, therefore the square on DC is quadruple of the square on CA, that is, CG is quadruple of AH.
But the gnomon MNO was also proved quadruple of AH; therefore the gnomon MNO is equal to CG.
And, since DC is double of CA, while DC is equal to CK, and AC to CH, therefore KB is also double of BH. [VI. 1]
But LH, HB are also double of HB; therefore KB is equal to LH, HB.
But the whole gnomon MNO was also proved equal to the whole CG; therefore the remainder HF is equal to BG.
And BG is the rectangle CD, DB, for CD is equal to DG; and HF is the square on CB; therefore the rectangle CD, DB is equal to the square on CB.
Therefore, as DC is to CB, so is CB to BD.
But DC is greater than CB; therefore CB is also greater than BD.
Therefore, when the straight line CD is cut in extreme and mean ratio, CB is the greater segment.
Therefore etc. Q. E. D.
LEMMA.
That the double of AC is greater than BC is to be proved thus.
If not, let BC be, if possible, double of CA.
Therefore the square on BC is quadruple of the square on CA; therefore the squares on BC, CA are five times the square on CA.
But, by hypothesis, the square on BA is also five times the square on CA; therefore the square on BA is equal to the squares on BC, CA: which is impossible. [II. 4]
Therefore CB is not double of AC.
Similarly we can prove that neither is a straight line less than CB double of CA; for the absurdity is much greater.
An XML version of this text is available for download,
with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted
changes, storing new additions in a versioning system.
Question: What is the relationship between CB and BD, according to the proof? Answer: CB is to BD as DC is to CB
Question: What is the relationship between KB and BH? Answer: KB is double of BH
Question: Can BC be double of AC? Answer: No, it cannot be double of AC
Question: What is the ratio in which CD is cut, according to the proposition? Answer: Extreme and mean ratio | 677.169 | 1 |
The student uses geometric concepts and procedures in a variety of situations.
3.1
The student recognizes geometric shapes and compares their properties in a variety of situations.
3.1.A1
solves real-world problems by applying the properties of (2.4.A1g):
3.1.A1A
plane figures (circles, squares, rectangles, triangles, ellipses, rhombi, parallelograms, hexagons, pentagons) and the line(s) of symmetry; e.g., twins are having a birthday party. The rectangular birthday cake is to be cut into two pieces of equal size and with the same shape. How would the cake be cut? Would the cut be a line of symmetry? How would you know?
3.1.A1B
solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases; e.g., ribbon is to be glued on all of the edges of a cube. If one edge measures 5 inches, how much ribbon is needed? If a letter was placed on each face, how many letters would be needed?
3.1.A1C
intersecting, parallel, and perpendicular lines; e.g., relate these terms to maps of city streets, bus routes, or walking paths. Which street is parallel to the street where the school is located?
length to the nearest eighth of an inch or to the nearest centimeter (2.4.A1a), e.g., in science, we are studying butterflies. What is the wingspan of each of the butterflies studied to the nearest eighth of an inch?
3.2.A1B
temperature to the nearest degree (2.4.A1a), e.g., what would the temperature be if it was a good day for swimming?
3.2.A1C
weight to the nearest whole unit (pounds, grams, nonstandard units) (2.4.A1a), e.g., if you bought 200 bricks (each one weighed 5 pounds), how much would the whole load of bricks weigh?
3.2.A1D
time including elapsed time (2.4.A1a), e.g., Bob left Wichita at 10:45 a.m. He arrived in Kansas city at 1:30. How long did it take Bob to travel to Kansas City?
3.2.A1E
hours in a day, days in a week, and days and weeks in a year (2.4.A1a), e.g., John spent 59 days in New York City. How many weeks did he stay in New York City?
3.2.A1F
months in a year and minutes in an hour (2.4.A1a), e.g., it took Susan 180 minutes to complete her homework assignment. How many hours did she spend doing homework?
3.2.A1G
Question: What are the two main types of geometric figures discussed? Answer: Plane figures and solids. | 677.169 | 1 |
The common flat rotation is done around the z-axis so z-values for the rotated points will remain unchanged but x- and -values may change. I said "may change" since a 360º rotation around any angle takes us to the same point as before.
To figure out how the x and y values change in a rotation around the z-axis we look at the two vectors (1,0,0) and (0,1,0). If we draw a circle in the center of our x,y-plane with the same radius as the distance to our points, we expect the points to move along the edge of this circle. We can easily imagine a counter-clockwise rotation of θ for both of these vectors and draw two new vectors that point to our expected end result. Basic trigonometry (sine and cosine) helps us express how the new points relate to the old points.
The transformed x-only-vector gives the values for the first column of our rotation matrix and the transformed y-only-vector gives us the values for the second column of our rotation matrix. The third or fourth columns don't alter the transformed vector so these are the same as for an identity matrix. The resulting rotation matrix is thus:×
x
y
z
1
=
cos θ⋅x
-
sin θ⋅y
sin θ⋅x
+
cos θ⋅y
z
1
The rotation matrix for a rotation around the z-axis.
To verify that this matrix works for a vector with both x and y components is left as an exercise for the reader. Pick a new vector with both x and y components and use the above matrix to calculate the rotated vector. Finally draw the rotated vector in a 2D plane to that the end result meets our expectations.
3D rotations and perspective
By applying the same techniques to rotations around the x-axis and y-axis we can figure out their rotation transforms (seen below).
Rx(θ) =
1
0
0
0
0
cos θ
-sin θ
0
0
sin θ
cos θ
0
0
0
0
1
Ry(θ) =
cos θ
0
sin θ
0
0
1
0
-sin θ
0
cos θ
0
0
0
0
1
Rz(θ) =The individual rotation matrices for all three axes.
While the point is correctly transformed in 3D space it doesn't look like a 3D rotation at all. This is because the 3D point is projected to the 2D screen without perspective. If you would go back and scale or translate the z-value you would experience the same problem (though there you would see no difference at all). This is not how we expect 3D objects to look. We expect objects far away to appear smaller and objects up close to appear bigger.
Question: Which values remain unchanged during a rotation around the z-axis? Answer: z-values | 677.169 | 1 |
In computer graphics, objects are often represented as triangulated polyhedra in which the vertices are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly, such as colors, reflectance properties, textures, and surface normals; these properties are used in rendering by a vertex shader, part of the vertex pipeline.
Mathematics
Vertex (geometry), a corner point of a polygon, polyhedron or general polytope, or by extension of a computer graphics point object.
.....A spatial point is a concept used to define an exact location in space. It has no volume, area or length, making it a zero dimensional object. Points are used in the basic language of geometry, physics, vector graphics (both 2D and 3D), and many other fields. .....Click the link for more information.
POLYGONE is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.
The range, also referred to as the Multi-national Aircrew Electronic Warfare Tactics Facility (MAEWTF), is .....Click the link for more information.
polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges.
The word polyhedron comes from the Classical Greek πολυεδρον, from poly- .....Click the link for more information.
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. Beyond that, the term is used for a variety of related mathematical concepts. .....Click the link for more information.
face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word hedra which means faceconvex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. .....Click the link for more information.
In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. A simple polygon has exactly one internal angle per vertextessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. .....Click the link for more information.
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (in modern language, which had better categorical properties), but still .....Click the link for more information.
Question: What is tessellation? Answer: A collection of plane figures that fills the plane with no overlaps and no gaps.
Question: What is the term for an angle formed by two sides of a simple polygon that share an endpoint? Answer: Interior angle | 677.169 | 1 |
Diagrams, week 6
This week, we were discussing definitions. In particular, we were trying to decide whether we needed to include the measures of the congruent angles in our definition of a rectangle, or whether simply stating that they are congruent would be good enough.
That is, can we define a rectangle as an equiangular quadrilateral? Or do we need to specify a quadrilateral with four 90° angles?
This led to discussion of whether it is possible for a quadrilateral to have all four of its angles obtuse.
One student argued "No". She argued that a rhombus typically has two obtuse angles and two acute angles; if you try to make the acute ones obtuse, the obtuse ones become acute.
The class accepted that argument (much to my chagrin).
So I stepped in, saying that I wasn't convinced. Specifically, I said:
You are arguing that a quadrilateral can't have four obtuse angles by showing that a rhombus can have at most two right angles. But I don't think that's true of quadrilaterals in general. I think it may be possible for a quadrilateral to have three obtuse angles.
If that's possible, then your argument doesn't show what we all seem to think it shows. And maybe four obtuse angles is possible.
A few minutes later, a student produced the diagram above on the board. A quadrilateral with three obtuse angles. Furthermore, we all agreed that we could imagine tweaking things so that all three of these angles would be congruent.
What have we learned? That we still don't know whether we need to state that the congruent angles in a rectangle are right.
3 Responses to Diagrams, week 6
Can you stress to your students that we like to have definitions which make sense not only in specific cases, but which can be used universally? For example, we are happy with the term "regular pentagon" to describe a specific 5-sided polygon, and that description would be universally understood Stating instead that we have a "pentagon where all angles are 108 degrees" is not only awkward, it comes off as a bit silly and superfluous. The fact that all angles in a regular quadrilateral are 90 degrees is not an essentially part of the definition, but can be instead characterized as a corollary of the definition.
Bob – I agree with you about the pentagon, but with rectangles we are fighting against two different things: students familiarity with rectangles and the importance and prominence of right angles. Of course, defining a rectangle as an equiangular quadrilateral has a certain elegance to it, but I think it will take a long time to lead students there. This is especially true since geometry is one of the first times students bump into the difference between the definition of a mathematical object and its properties.
Question: What was the main point of Bob's response? Answer: Bob agreed that a regular pentagon is universally understood, but defining a rectangle as an equiangular quadrilateral might take time due to students' familiarity with rectangles and the importance of right angles.
Question: What is the universal understanding of a regular pentagon, according to Bob? Answer: A regular pentagon is universally understood as a specific 5-sided polygon. | 677.169 | 1 |
To construct a tetrahedron, simply insert the right-hand projection of one unit into the left-hand pocket of another. Now, add a third unit to join the first two, forming one of the tetrahedral frame's four points and three of its six edges. Use the remaining three units to complete the tetrahedron.
Dodecahedron
Now, the tricky part is weaving your five tetrahedra together to form a dodecahedron. The rule of thumb is that the peak of each tetrahedron should come through the base of another – it's also helpful to keep in mind that the 20 points of the combined tetrahedra form the pentagonal points of the dodecahedron. The diagrams described above are especially helpful in assembling the final platonic solid, but the peak-base rule can also be used to successfully weave the dodecahedron.
Wrap-Up Questions
Can you make any other platonic solids using the modular units that form the tetrahedra?
Why is the 60° angle important? Could you complete this model with units formed by any other angles?
Could thinner units be made with the 60° angles intact?
Why does the method used to form the 60° angle in the construction of the basic units work? (Hint: Check out Huzita's fifth axiom.)
About Author
Maria Rainier is a freelance writer and blog junkie. She is currently a resident blogger at First in Education, where recently she's been researching online mechanical engineering degrees and blogging about student life. In her spare time, she enjoys square-foot gardening, swimming, and avoiding her laptop.
Question: Who is the author of the text? Answer: Maria Rainier
Question: What is one of Maria Rainier's interests outside of writing? Answer: Square-foot gardening | 677.169 | 1 |
In response to question 2 (is the answer 3?). If that is the right answer, I drew a triangle A, top D, and E with B as AD bisector and C as base bisector then formed the rest of the internal triangle that began with the line BC. So now we have an upside down triangle inscribed in the right side up triangle, forming 4 separate small triangles. In my eyes...ABC looks like 1/4 of the area of 12..or 3. If that's not the answer...I'm stumped too.
If what you say is true...and X and 10 have no similar prime factors, then how can the answer be 9? The question asks which # can X be a multiple of. If X is a multiple of 9, then it can be 9 and then the disimilar factorization holds true but what if X is 18 (a multiple of 9)..then 18 and 10 have 2 as a common prime factor. I don't understand. Unless relatively prime natural numbers only include common even prime factorization? Help! Now it's become my question..thanks
Question: What shape is described as being formed by the lines BC? Answer: An upside down triangle | 677.169 | 1 |
Coincidence is easily checked by testing if a point on one line, say P0, also lies on the other line Q(t). That is, there exists a number t0 such that: P0= Q(t0) = Q0+ t0v. If w = (wi) = P0 – Q0, then this is equivalent to the condition that w = t0v for some t0 which is the same as for all i. In 2D, this is another perp product condition: . If this condition holds, one has , and the infinite lines are coincident. And if one line (but not the other) is a finite segment, then it is the coincident intersection. However, if both lines are finite segments, then they may (or may not) overlap. In this case, solve for t0 and t1 such that P0= Q(t0) and P1= Q(t1). If the segment intervals [t0,t1] and [0,1] are disjoint, there is no intersection. Otherwise, intersect the intervals (using max and min operations) to get . Then the intersection segment is . This works in any dimension.
Non-Parallel Lines
When the two lines or segments are not parallel, they might intersect in a unique point. In 2D Euclidean space, infinite lines always intersect. In higher dimensions they usually miss each other and do not intersect. But if they intersect, then their linear projections onto a 2D plane will also intersect. So, one can simply restrict to two coordinates, for which u and v are not parallel, compute the 2D intersection point I at P(sI) and Q(tI) for those two coordinates, and then test if P(sI) = Q(tI) for all coordinates. To compute the 2D intersection point, consider the two lines and the associated vectors in the diagram:
To determine sI, we have the vector equality where . At the intersection, the vector is perpendicular to , and this is equivalent to the perp product condition that . Solving this equation, we get:
Note that the denominator only when the lines are parallel as previously discussed. Similarly, solving for Q(tI), we get:
The denominators are the same up to sign, since , and should only be computed once if we want to know both sI and tI. However, knowing either is enough to get the intersection point I = P(sI) = Q(tI).
Further, if one of the two lines is a finite segment (or a ray), say P0P1, then the intersect point is in the segment only when (or for a ray). If both lines are segments, then both solution parameters, sI and tI, must be in the [0,1] interval for the segments to intersect. Although this sounds simple enough, the code for the intersection of two segments is a bit delicate since many special cases need to be checked (see our implementation intersect2D_2Segments()).
Plane Intersections
Line-Plane Intersection
Question: What is the relationship between the lines in 2D Euclidean space when they are not parallel? Answer: They always intersect.
Question: What is the condition for the intersection point I to lie within a finite segment P0P1? Answer: 0 ≤ sI ≤ 1
Question: What is the method to find the intersection point of two lines in higher dimensions? Answer: Restrict to two coordinates, compute the 2D intersection point, and then test if the intersection point lies on both lines for all coordinates.
Question: In higher dimensions, what usually happens when two lines or segments are not parallel? Answer: They usually miss each other and do not intersect. | 677.169 | 1 |
The cross product between two lines give the intersection point: x = cross(l,m).
Intersection of two lines (l and m) is a point. The intersection point x is in the line l: dot(x,l) = 0. Also, the point x is in the line m: dot(x,m) = 0. Therefore, the cross product of l, m gives the intersection point x.
Uncertainty and geometric constraint
Uncertainty of geometric entity
Uncertainty of a point?
Uncertainty of a line?
Uncertainty of a direction?
Uncertainty of a plane?
An example: triangle
3 angles of triangle
Imagine that you are measuring three angles of a triangle with a protractor. From the first trial, you got values [30.5, 60.0, 90.1] degrees. Which are not perfect because the summation of them is not equal to 180! So, you have decided to measure angles 10 times and average them. However, still you are not sure that the sum of the averages will be 180.
Dual Quaternions
"Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics, and in applications to 3D computer graphics, robotics and computer vision." Wiki
Working on the manifold
Registration of multiple views
Point Cloud Only
Point registration problem (1) finds point-to-point correspondences and (2) estimates the transformation between them. Previous methods can be divided into three categories: the ICP (Iterative Closest Point), soft assignment methods, and probabilistic methods.
ICP can be trapped in local minima and sensitive to initialization and the threshold to accept a match. Hence, the nearest point strategy is replaced by soft assignments within a continuous optimization framework. But, the convergence property is not guaranteed in the presence of outliers. The point-to-point assignment problem can be cast into estimation of parameters of a (Gaussian) mixture. --> EM-ICP
ICP, EM-ICP
Tensor
Initialization
Source codes
RGB 2D features + depth values
Since we have both RGB and depth map, registration of two views can be done by simple 3D feature matching. Given two sets of corresponding 3D feature descriptors, X = [x1, x2, ..., xn] and Y = [y1, y2, ..., yn], the rigid motion between X and Y, Y = T X, is computed directly. A robust estimation method such as RANSAC provide accurate results in the presence of outliers.
2D, 2.5D and 3D features
Curvature estimation: HK Segmentation
See HK Segmentation
Curvature estimation in the presence of noise depth map
2.5D Features
Using 2D image features on a depth map
Question: What is the process of registering two views using RGB 2D features and depth values? Answer: Given two sets of corresponding 3D feature descriptors, X and Y, the rigid motion between them, Y = T X, is computed directly. A robust estimation method like RANSAC is used to provide accurate results in the presence of outliers. | 677.169 | 1 |
Remember to analyze your incorrect questions from Veritas Prep's question bank. Use this data to understand your strengths and weaknesses and focus your GMAT prep on the area that need it most!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing tips and tricks to help students better prepare for the GMAT and the SAT.
]]> of Geometry - Part I
13 May 2013 16:03:46 +0000Karishma continue with geometry today. We would like to discuss how drawing extreme diagrams can help you solve questions. Most GMAT questions are quite intuitive and hence our non-traditional methods are perfect for them. They are not typical MATH problems per se; instead, they are logical puzzles. If you can prove why some things will not work, it means whatever is left will work.
Let me explain with the help of an official Data Sufficiency question.
Question:
In the figure above, is the area of the triangle ABC equal to the area of the triangle ADB?
Statement 1: (AC)^2=2(AD)^2
Statement 2: ∆ABC is isosceles.
Solution:
When presented with this question, people see right triangles and jump to Pythagorean theorem, isosceles triangles and then wage a war on AC, AB, CB and AD relations. Well, that is our traditional approach. But what do we do if making equations and solving for relations isn't our style?
We make diagrams and figure out the relations! One thing that is apparent the moment we read statement 1 is that the figure is not to scale. From the figure it looks as if AD is greater than or at best, equal to AC. That itself is an indication that if you draw the figure on your own, you could see something that will make this question very simple. The question setter doesn't want to show you that and hence he made the distorted figure.
Anyway, let's first analyze the question. Then we will look at the statements.
We need to compare areas of ABC and ADB. Both are right angled triangles.
Area of ABC = (1/2) * AC * BC
Area of ADB = (1/2) * AD * AB
We need to figure out whether these two are the same.
Think about it this way – we are given a triangle ABC with a particular area. So the length of AD must be defined. If AD is very small, (shown by the dotted lines in the diagram given below) the area of ADB will be very close to 0. If AD is very large, the area will be much larger than the area of ABC. So for only one value of AD, the area of DAB will be equal to the area of ABC.
We need to figure out whether for the given relations, the triangles have equal area.
Statement 1: (AC)^2=2(AD)^2
Question: According to the text, if AD is very small, what is the area of triangle ADB?
Answer: Very close to 0
Question: What is one of the non-traditional methods mentioned in the text to solve GMAT questions?
Answer: Drawing extreme diagrams.
Question: Which statement is being analyzed in the text: Statement 1 or Statement 2?
Answer: Statement 1: (AC)^2=2(AD)^2
Question: Which of the following is NOT a traditional approach to solving the given problem?
A) Using Pythagorean theorem
B) Making equations and solving for relations
C) Drawing diagrams and figuring out relations
Answer: C) Drawing diagrams and figuring out relations (as it's mentioned as a non-traditional method) | 677.169 | 1 |
Information about vertex (geometry)
Definition
For each of those figures, a vertex is a point formed by the intersection of faces of the object: a vertex of a polygon is the point of intersection of two polygon edges, a vertex of a polyhedron is the point of intersection of three or more polyhedron facets, and a vertex of a d-dimensional polytope is the intersection point of d or more polytope facets. A vertex can also refer to an angle, the point where two rays begin or meet, where two line segments join or meet, where two lines cross (intersect), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place.
In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π; otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and concave otherwise.
A vertex of a plane tessellation is a point where three or more tiles meet; generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.
Geometric vertices are related to vertices of graphs, in that the 1-skeleton of a polyhedron or polytope is a graph, the vertices of which correspond to the vertices of the polyhedron or polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.
Principal vertex
A polygon vertex of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at and .
There are two types of principal vertices, ears and mouths.
Ears
A principal vertex of a simple polygon P is called an ear if the diagonal that bridges lies entirely in P. (see also convex polygon)
Mouths
A principal vertex of a simple polygon P is called a mouth if the diagonal if the interior of lies in the outside the boundary of P). (see also concave polygon)
Vertices in computer graphics
Question: How many edges meet at a convex vertex of a polygon? Answer: Two edges meet at a convex vertex of a polygon.
Question: What is a mouth in a simple polygon? Answer: A mouth is a principal vertex in a simple polygon where the interior of the diagonal lies outside the boundary of the polygon.
Question: What is the connection between geometric vertices and the vertices of a graph? Answer: The 1-skeleton of a polyhedron or polytope is a graph, and the vertices of the graph correspond to the vertices of the polyhedron or polytope. | 677.169 | 1 |
My Three Triangle Puzzle
What do these three triangles have in common, besides a side of
seven? You might want to think about it before you go on to the next
paragraph.
Comments: Unless you recognize the second and third triangles,
the above question seems a little difficult to answer. The areas are all
different, and the perimeters are all different. Most of the angles are
different. Could the upper angle in all three triangles be the same (60
degrees)? It looks like that might be the answer, but how do we prove that?
The Pythagorean Theorem comes to mind
(let's ignore the law of cosines for now). How could we possibly apply the
Pythagorean theorem?
If those angles are 60 degrees, then making a copy of
the middle triangle, next to the rightmost triangle, would produce an
equilateral triangle. See the drawing on the right. It looks like a triangle.
Can we prove that the right side is a straight line? This is the same puzzle as
the above, but from a different perspective. Does that help us any?
Solution: I guess the easiest way of solving
our problem is to look at the equilateral triangle on the left, and show that
the two appropriate triangles are congruent with those in the previous diagram.
By the Pythagorean Theorem, the altitude (h) of the equilateral triangle is
4sqr(3) (4 times the square root of 3). A further application of the
Pythagorean Theorem shows that the hypotenuse (x) of the skinny right triangle
(with sides h, 1, and x) is indeed 7. And so the two appropriate triangles are
congruent in the two diagrams (by SSS (see Congruence Of Triangles, Part I)). The
appropriate supplementary angles are also congruent. And so the rightmost line
in the upper right diagram is indeed a straight line. And incidentally, the
angles that we thought might be 60 degrees are indeed 60 degrees, which answers
our original question.
Above, I mentioned the law of cosines, which is that in any triangle
ABC, c^2=a^2+b^2-2ab(cos(C)) (with c^2 meaning c squared). The cosine of 60
degrees is 1/2. We could have applied this formula to the original triangles,
and shown that the cosine of the upper angles was 1/2 in all three triangles.
That would have been too easy.
One further diagram, I've redrawn all three triangles
on the same base. And we see that all five vertices are on a circle. This is
the result of one of Euclid's theorems: An angle inscribed in a circle is half
the central angle of the inscribed arc. So all inscribed angles with the same
arc are congruent angles.
On the left, we see some more triangles
that have integer-length sides. The leftmost angle is 60 degrees (there is an
equilateral triangle in the diagram).
Question: According to Euclid's theorem, what is the relationship between an inscribed angle and its central angle? Answer: An inscribed angle is half the central angle of the inscribed arc.
Question: What is the length of the altitude (h) of the equilateral triangle on the left? Answer: 4 times the square root of 3 (4√3).
Question: Which congruence rule is used to prove that the two appropriate triangles are congruent? Answer: SSS (Side-Side-Side).
Question: Can the areas and perimeters of the three triangles be the same? Answer: No, they cannot. | 677.169 | 1 |
Ratios in Right Triangles
In this lesson our instructor talks about ratios in right triangles. First she talks about trigonometric ratios of sine, cosine, and tangent. Then she discusses trig function, and inverse trig functions. She finishes with a lecture on SOHCAHTOA. Four extra example videos round up this lesson.
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Ratios in Right Triangles
Trigonometry: The study of involving triangle measurement
sine (sin) = opposite/hypotenuse
cosine (cos) = adjacent/hypotenuse
tangent (tan) = opposite/adjacent
SOHCAHTOA
Ratios in Right Triangles
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Question: What does the acronym SOHCAHTOA stand for? Answer: SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent) | 677.169 | 1 |
Sine- Used to find the OPPOSITE or HYPOTENUSE. A good way to remember this is the acronym, SOH.
Cosine- Used to find the ADJACENT or HYPOTENUSE. A good way to remember this is the acronym, CAH.
Tangent- Used to find the OPPOSITE or ADJACENT. A good way to remember this is the acronym, TOA.
It can be difficult to remember all this, but if you follow SOH CAH TOA it will make it a lot easier!
This week we learned all about areas in geometry class. In fact, we just took a test on them. Very early on we learn the basic ones: rectangle, square, triangle. However, there are many more formulas to finding areas that I have just learned!
Here's a list:
My favorite area to find is the rhombus. This is because I never knew how to do it before just a couple days ago. I find it fascinating to learn new things! Also, there is sometimes algebra involved, which I really enjoyI'm sure many of you have heard of the Pythagorean theorem, but do you really know what it is? If not, I am here to tell you! This theorem is used to find the area of triangles, where each side corresponds with a letter (A, B, or C). It is as follows:
A squared * B squared= C squared.
Although I learned this last year in algebra, it was good to refresh my memory. Also, this year we are doing more complex things with it, such as finding the area of other shapes. Some of these other shapes include trapezoids, rhombuses, and kites. When it is used in these shapes, it helps find one of the side or diagonal lengths. I really enjoy using this because it is an equation!
This past week we jumped back into algebra by refreshing our minds on radicals! Put simply, radicals are square roots. Right now we are working on simplifying them into simplest radical form. This is when they can no longer be broken down any further. Ms. Jovanovich has taught us that the best way to do this is to use prime factorization!
Although just refreshing now, next week we will apply radicals to geometry. We will be using them to find the area of shapes.
I'm very excited to start this unit because I feel more comfortable with algebra concepts than I do with geometry. I'm interested in seeing how I like radicals when applied to geometry! Hopefully they'll be just as fun as they were in algebra!
This past week we continued to learn about transformations! However, this time we were focusing on reflections. In the past we have learned about these, but this time it was more complex. At first I thought they were a little difficult, but now I'm more confident with them.
There are two things you need for a reflection:
1. A preimage
2. A line of reflection
You then reflect the points to the opposite side of the line of reflection. Make sure that they are on a line perpendicular to the line of reflection.
Question: What is the Pythagorean theorem used for? Answer: The Pythagorean theorem is used to find the area of triangles and to calculate the length of one side of a right-angled triangle when the lengths of the other two sides are known.
Question: What does SOH stand for in the context of trigonometry? Answer: SOH stands for Sine - Opposite - Hypotenuse.
Question: Which trigonometric function is used to find the adjacent side in a right-angled triangle? Answer: Cosine
Question: Which shape's area calculation did the user find fascinating to learn recently? Answer: Rhombus | 677.169 | 1 |
The 0 on the vernier scale is spaced the distance of exactly one r u l e r m a r k (in this case, one tenth of a n inch) from the left hand end of the vernier. Therefore the 0 is a t a position between r u l e r m a r k s which is comparable to the position of the end of the bar. In other words, the 0 on the vernier i s about halfway between two adjacent m a r k s on the r u l e r , just as the end of the bar i s about halfway between two adjacent m a r k s . The 1 on the vernier scale i s a little c l o s e r to alinement with a n adjacent r u l e r mark; in fact, it i s one hundredth of a n inch c l o s e r to alinement than the 0. This is because each space on the vernier i s one hundredth of a n inch s h o r t e r than each space on the r u l e r . Each successive m a r k on the vernier scale i s one hundredth of a n inch closer to alinement than the preceding m a r k , until finally alinement is achieved a t the 5 mark. This means that the 0 on the vernier must be five hundredths of a n inch f r o m the n e a r e s t r u l e r mark, since five increments, each one hundredth of an inch in s i z e , w e r e used before a m a r k was found in alinement. We conclude that the end of the bar i s five hundredths of a n inch from the 2.9 m a r k on the r u l e r , since its position between m a r k s i s exactly comparable to that of the 0 on the vernier scale. Thus the value of our measurement is 2.95 inches. The foregoing example could be followed through for any distance between markings. Suppose the 0 m a r k fell seven tenths of the distance between r u l e r markings. It would take
seven vernier markings, a loss of one-hundredth of an inch each time, to bring the marks in line a t 7 on the vernier. The vernier principle may be used to get fine linear readings, angular readings, etc. The principle i s always the same. The vernier has one more marking than the number of markings on an equal space of the conventional scale of the measuring instrument. For example, the vernier caliper (fig. 6-5) has 25 markings on the vernier for 24 on the caliper scale. The caliper is marked off to read to fortieths (0.025) of an inch, and the vernier extends the accuracy to a thousandth of an inch.
Figure 6-5.-A vernier caliper. Vernier Micrometer
Question: How many increments of one hundredth of an inch are there between the 0 on the vernier and the nearest ruler mark? Answer: Five.
Question: What is the value of the measurement when using the vernier principle? Answer: 2.95 inches.
Question: What is the measurement of the bar's end from the 2.9 mark on the ruler? Answer: Five hundredths of an inch.
Question: How much closer is the 1 on the vernier scale to alignment with an adjacent ruler mark compared to the 0? Answer: One hundredth of an inch. | 677.169 | 1 |
Solution to puzzle 86: Folded card
A piece of card has the shape of a triangle, ABC, with BCA a right angle. It is folded once so that:
C coincides with C', which lies on AB; and
the crease extends from Y on BC to X on AC.
If BC = 115 and AC = 236, find the minimum possible value of the area of YXC'.
We will pursue the general case as far as is practicable.
Assign Cartesian coordinates to the vertices of the triangle: C = (0,0),B = (0,a),A = (b,0), and Y = (0,r),X = (s,0).
Let the equation of line CC' be y = kx, for some k > 0, such that Y lies on BC and X lies on AC. We will express the area of YXC' in terms of a, b, and k.
The equation of line AB is ax + by = ab.
Hence C' = (ab/(a + bk), abk/(a + bk)).
It is easy to verify that, if C' is moved closer to B (so that X coincides with A), or if C' is moved closer to A (so that Y coincides with B), the area of YXC' exceeds the value of w, calculated above.
Since w is a continuous function, and has only one turning point for k > 0, it follows that k = 5/4 represents a minimum.
Therefore the minimum possible value of the area of YXC' is 1841449/640 = 2877.2640625 sq. units.
Remarks
An expression such as 236k3 + 345k2 − 708k − 115 cannot easily be factored by inspection! Although the general cubic equation can be solved analytically by hand (see references below), this can be quite a lengthy process. Fortunately, even if you don't own a mathematical software package, there are a number of online calculators available, such as the QuickMath Equation Solver. For a cubic equation with rational coefficients, this solver will provide exact solutions in terms of (possibly complex) radicals, as well as approximate numerical solutions.
Question: What is the value of k that represents the minimum area? Answer: k = 5/4
Question: What is the shape of the card in the puzzle? Answer: A triangle
Question: What is the equation of line AB? Answer: ax + by = ab | 677.169 | 1 |
The diagram you drew is considered a hexagon but often the way mathematics grows is when an intuitive/familiar concept evolves to get insight into a broader collection of ideas. This has been especially true for geometry. The critical feature for a polygon is that straight line segments join up the the points involved. Early mentions of polygons implicitly assumed that one was dealing with convex polygons, one had distinct points, one had no three of the points on a straight line or no three consecutive points on a straight line, that different points could not sit on top of each other, and that the points all were located in a single plane. Now geometers often investigate situations where some of these conditions are loosened and as a result many exciting new geometrical phenomena have come to light.
For a fascinating look at this type of issue for polyhedra instead of polygons look at the wonderful paper of Branko Grünbaum, "Are your polyhedra the same as my polyhedra?"
Check your kid's geometry syllabus: it may be he's being taught simple, basic geometry and thus n-gons are usually considered to be convex, i.e.: the interior (and perimeter, too) of the n-gon must contain the whole line segment connecting any two points in the interior. Under this agreement what you drew wouldn't qualify as hexagon or, perhaps more accurately, convex hexagon.
Yes, It is Considered as a Hexagon. There is a difference between an Irregular Hexagon and a Regular Hexagon. A regular hexagon has sides that are segments of straight lines that are all equal in length. The interior angles are all equal with 120 degrees. An irregular hexagon has sides that may be of different lengths. It also follows that the interior angles are not all equal. Some interior angles may be greater than 180 degrees, but the sum of all interior angles is 720 degrees. Hope this gives u an idea about it.
For polygon the closing sides from start to end point after the desired number reached is said to be a polygon. Specific term come into action only when the condition like angle and length are taken into account....
say for example, i am dreaming to draw 6 sides closed (still its too a polygon with 6 sides)
I am drawing a circle, its total angle is 360.
Now i am planning to divide its total angle into 6 parts(sectors) I have made the imaginary lines for the angles inscribed. 360/6 = 60 degree
now i am joining the lines starting from center of circle to the circumference of it until i'm touching it.
now i am marking the points in which formed on the circumference of the circle due to the line joining from the center of circle to the circumference due to the angle i marked (60 degree)
Now join the marked points each other remove the circle you will get a perfect HEXAGON
The above shown image is of irregular hexagon with uneven angles and edges.
Below shown image is for regular hexagon with even edges and angles Image.
so you can divide hexagons into 2 subcategories.
1. even and 2. uneven.
Question: Is the diagram drawn in the text considered a hexagon? Answer: Yes, it is considered a hexagon. | 677.169 | 1 |
Polar Coordinates Game
The game helps students to understand polar coodinates.
A point in polar coordinates is defined by an angle[0° ,360°) and distance from a fixed point
(center of the polar coordinate system).
Game Instructions:
You have to locate the ship on the map in polar coordinates: set an angle and distance by the sliders on the right.
When the ship is located the game displays: "Magnifique".
Press reload button in the top right corner to start a new game.
Question: What is the second parameter used to define a point in polar coordinates? Answer: Distance from the center | 677.169 | 1 |
i just made these because that the last time i was working on a 3d game, i was wondering how to do 3d distance and i was making everything all complicated and stuff but its really not that hard. ill try to make a picture that demonstrates exactly how all of it works if some of you do not get it... pythagorian theorum and point distance 2d pretty much the same thing but its just two different ways to go about finding the distance between two 2d points. although pyth thor has less arguments, it needs you to have the distance of two parts already so i couldnt say which one is better or anything and that is why i made them both. i give permission for everyone to use these and not have to give me any credit if you dont want to.
EDIT: ty for the spelling correction and "var" tip! btw, "pyth" and "pt dist" do different things. pt_dist_2d gets x and y values for 2 different points, finds the distance between the two x's and y's and uses the pythagorian theorum formula to find the distance while pth_theor gets 2 lengths, meaning u already have a variable or something with the length and it just uses pythagorian theorum formula on them to get the distance. so the pythagorian theorem is used for distance when u know the length between the x's and y's already and pt_dist_2d is if you only know the points at which you want to find the distance between. like i mentioned above, they are pretty much the same thing, but there are differences...
Question: Can I use the provided functions without giving credit? Answer: Yes, the author gives permission for everyone to use these functions without giving credit.
Question: Which function is used when you already know the lengths between the x's and y's? Answer: 'pth_theor' | 677.169 | 1 |
Jon's Answer:
Gavin...are the measures to your triangle in cm's If they are...then the formula you should commit to memory...write it on your hand...or something....is area(of a RIGHT ONLY triangle!!) = one-half base times height[in words] but in an equation...a = ½(b × h), If its not a RIGHT TRIANGLE...then its another formula. Those you should commit to memory as well...write them on your pillow(if your mom let's you).
And another question...if your measures aren't in 'cm'...then you should know how to change whatever measures you have...ie...inches...feet...meters...miles...whatever...be able to convert them to cm's.
I don't really know how to explain but basicly i have the exterior angel of 25 degrees on the inside which happens to be on the same side and has the diagonals says 10 degrees on top and the variable V on the bottom . On the side with no diagonals says 155 degrees then under that says the variable C.C also happens to be apart of the diagonals the other angel has nothing because we don't. Need to answer it.
Jon's Answer:
Jourdan...I'm limited by the functions of this e-mail...so I'm going to describe it to you, so that you and put a visual picture and it will help answer your own question. I'm going to describe it to you, step by step, and if you will draw it. Jourdan...I'm assuming you know the definitiond of some of the words I use.
Step 1
On your sheet of paper, draw to parallel lines...(remember the definition of two parallel lines)...now that you have drawn the lines...draw a third line which intersects with the two lines...its gonna look like a railroad track with a telephone pole that fell across it at an angle and touching both lines at some point. Not at a right angle, or straight across.
Step 2
This configuration creates 8 angles, 4 on one line and 4 on the other line, correct? Can you see that? this is where the rules and axioms come into play which defines 2 parallel lines.
Step 3
Pick one of the lines that intersect, with 4 angles. If the lines are not right angles, you will have one group of angles, one larger than the other... that's if the lines do not create right angles. Look under the parallel line, you will see similar lines, that are upside down and reversed.
Step 4
The point where the two lines intersect, the vertex, one of the angles is
larger than the other. BUT...But...together, they create a straight line. Let's say one angle is Angle A, the larger angle, and the other angle, which is smaller, Angle B.(notice I have not used any measurement, like degrees!!!) SO...now we have the equation below.
Angle A + Angle B = a straight line RIGHT!!!!
Step 5
Question: If the triangle is not a right triangle, what other formulas should be committed to memory? Answer: The formulas for non-right triangles should also be committed to memory.
Question: What is the formula to calculate the area of a right triangle? Answer: The formula is a = ½(b × h), where 'a' is the area, 'b' is the base, and 'h' is the height.
Question: What is the measure of the angle on the side with no diagonals? Answer: The measure of the angle is 155 degrees.
Question: What is the relationship between the two angles at the point where the two lines intersect in the described figure? Answer: Together, they create a straight line, and the equation is Angle A + Angle B = 180 degrees. | 677.169 | 1 |
Look at where the lines intersect. There are 4 angles, RIGHT! We said Angle A was the larger angle. NOTICE...right across Angle A is another angle with is very similar. The angle is right across and under Angle B. Let's label that Angle D.
Step 6
Since Angle D is under Angle B notice that being next to one another they create a straight line...created by the intersection of one line to another. Since Angle B and Angle C create a straight line. So we can say:
Angle B + Angle D = a straight line RIGHT!! See that !!!
Step 7
But remember: Angle A + Angle B = a straight line right!!
But we also saide Angle B + Angle D = a straight line RIGHT!!!!
Do you see what happened here.... we can say Angle B = Angle B Right!!!
If this is true Angle A = Angle A and Angle D = Angle D RIGHT!!
But we also said
Angle A + Angle B = a straight line a nd Angle B + Angle D = a straight line
And since a straight line is going to equal a straight line....we can say
Angle A + Angle B = Angle B + Angle D
And since Angle B = Angle B....we can then say:
Angle A + Angle B = Angle B + Angle D so
Angle A = Angle D
With all of this said...now but in the degrees of the angles created by intersecting two parallel lines. Draw the two lines, put the degree marks in and complete the addition as about. We add because a line is 180*(degrees)
CORRECT!!!! THEN WE CAN SAY:
Angle A + Angle B = 180*
Angle B + Angle D = 180*
So if Angle A is 100* what is Angle B?
Angle A + Angle B = 180* Angle A = 100*
100* + Angle B = 180* SO!!!!
Angle B = 180* - 100*
Angle B = 80* Right!!!
Now...put in your values for your angles and see if they work out that way.
Good luck..work..using the concepts outlined..then you will be the MASTER!!
Jon's Answer:
Ashley...when you came across this problem...did you know what the solution would look like? I'm assuming that you knew what the problem was requiring you to do was find the values, that when multiplied together would give you the equation. x^2 - 5x + 6
The problem is asking you to find out what values, in this case, binomials, that when multipled together gives you the trinomial x^2 - 5x + 6
So what you're doing is dividing...in this case...finding the factors of the trinomial.. x^2 - 5x + 6
With that said, I'm assuming that you know your solution would be like:
Question: What is the equation that represents the sum of angles formed by intersecting parallel lines? Answer: Angle A + Angle B = Angle B + Angle D = 180 degrees
Question: Which angle is mentioned to be the larger angle initially? Answer: Angle A | 677.169 | 1 |
KaleidoHedron. The symmetries of an icosahedron are
used to replicate a pattern across the surface of a sphere.
What are the symmetries of an icosahedron? Pick any pair of adjoining vertices,
and call them v1 and v2. You can rotate the icosahedron so that any of its
12 vertices ends up at the original position of v1, and any of the 5 adjoining
vertices ends up at the original position of v2. That gives 60 distinct rotations.
An optional reflection, in the plane passing through the icosahedron's centre and
the original positions of v1 and v2, yields a total of 120 symmetries.
Each of these symmetries carries a different triangular piece of the
icosahedron's surface (one sixth of a triangular face) into the triangle bounded
by (the original positions of) v1, the point halfway between v1 and v2, and the
centre of one of the faces that has v1 and v2 as corners. All the symmetries can
be generated by repeated reflections in the three planes that pass through the
centre of the icosahedron and any of the three edges of this triangle.
Question: What is the total number of symmetries of an icosahedron? Answer: 120 | 677.169 | 1 |
CA Geometry: 15-16, similar triangles
Transcript
We're on problem 15, it asks us, if triangle ABC and triangle XYZ or two triangles such that, okay let me draw these two triangles. So, triangle ABC, maybe it look something like that. ABC and then we have triangle XYZ and we want to prove that they are similar. So, similar means that they look the same, they have the similar shape but they could be of different sizes so essentially all of their angles are the same, all the ratios of their side are the same so XYZ might just be a smaller version maybe I should have drawn a bigger one, XYZ. It looks like they want us to prove the triangles are similar. They tell us that side, the ratio of AB to XY, let me color that. They said the ratio of AB to XY is equal to the ratio of BC to YZ.
Which of the following would be sufficient to prove that triangles are similar? So, there's a couple of times when you know that a triangle is similar. Is it the ratio of all the sides are equal? Think about just giving two sides isn't enough because eventhough I drew them so that they are similar, it could look something like this. Maybe the ratio here is, I don't know two to one. So, maybe AB and XY maybe that's a two to one, maybe that's two and that's one like that, right. The ratio is 2 to 1 and maybe BC to YZ is also 2 to 1. For y, there's nothing to tell us that it can't open up like this. It could be like a two and a one ratio right or actually even better I don't want to draw them similar anymore, I want to draw them unsimilar. Maybe BC comes in like that, maybe YZ goes out like that and so just knowing that the ratio of two of the triangles are two of the sides are the same, that alone doesn't tell you that you're dealing with the similar triangle. These two triangles would be very different. These are definitely not similar triangles. In fact all of the angles would be different.
Now, how do you prove that something is similar? Well, actually I attempted this problem about two minutes ago and it stumped me because I think if I'm right that there is a mistake in the problem. In geometry class, they always teach you that if you know that the ratio of two of the sides are equal and that the angle between them are equal then that's sufficient to say that it is a similar triangle or that you can make the conclusion that the ratio of AC to XZ would also be equal. So, that's the answer they're probably looking for that if you have the ratio of AB to XY, you have the ratio of BC to YZ then you want the angle in between them to say okay these are similar triangles so you have to know that angle B is congruent to angle Y and that is choice B.
Question: Can the given information alone prove that the triangles are similar? Answer: No, the given information alone is not sufficient to prove that the triangles are similar.
Question: Which option in the transcript is the correct answer to the problem? Answer: Choice B: "If you have the ratio of AB to XY, you have the ratio of BC to YZ then you want the angle in between them to say okay these are similar triangles so you have to know that angle B is congruent to angle Y" | 677.169 | 1 |
Angles/323897: The measure of an angle is six more than twice the measure of its complement. Find the measures of both angles. 1 solutions Answer 231783 by nyc_function(2733) on 2010-07-19 23:52:08 (Show Source):
Graphs/323879: How many liters of a 40%-alcohol solution must be mixed with 10 liters of a solution that is 80% alcohol to get a solution that is 60% alcohol? 1 solutions Answer 231782 by nyc_function(2733) on 2010-07-19 23:50:19 (Show Source):
You can put this solution on YOUR website! I know that this is a geometry question but it can be solved using trigonometry.
We use the law of sines.
The sum of the angles of a triangle is 180 degrees.
To find the missing angle, add 51 + 75 and then subtract the sum from 180.
This will give you 54 degrees.
We now use the law of sines.
Let x = the missing side we want to find
sin51/x = sin75/25
x(sin75) = sin51(25)
x(sin75) = 19.42864904
x = 19.42864904/sin75
x = 20.11401756
The side opposite the smallest angle is 20 inches.
You can put this solution on YOUR website! Equilateral means all sides are the same.
The bottom side can be divided into two parts of equal length: 10cm = 5cm + 5cm.
Let h = height
Use the Pythagorean Theorem to find h.
h^2 + 5^2 = 10^2
Can you take it from here?
Mixture_Word_Problems/322471: The owner of Nuts2U Snack Shack mixes cashews worth $6.25 a pound with peanuts worth $2.00 a pound to get a half-pound mixed nut bag worth $1.70. How much of each kind of nut is included in the mixed bag? 1 solutions Answer 230848 by nyc_function(2733) on 2010-07-14 16:16:14 (Show Source):
You can put this solution on YOUR website! Let c = number of cashews
Let p = number of peanuts
c + p = 0.5
6.25c + 2.00p = 1.70
This is your system of equations in two variables.
Question: What is the missing side of the triangle in the geometry problem? Answer: 20.11401756 inches
Question: What is the measure of the complement of the angle with a measure of 75 degrees? Answer: 51 degrees
Question: What is the height (h) of the equilateral triangle with a side length of 10 cm? Answer: 8.66 cm (using h = √(10^2 - (5^2))) | 677.169 | 1 |
In summary, the equation means a point that is both (a) on the plane in which triangle A lies and is (b) on a ray in which the edge of a 2nd triangle B lies.
I thought that in 4x4 matrices a tranlation was possible through multiplication... but in truth I haven't even got an negation function.... Nor had I thought about them....
Well, things are more complicated. Assume a 2D space in which to do transformations. Assume that you have a 2x2 matrix R for rotation and a 2 vector t for translation. Using row vectors, a transformed point P' then is computed from its original point P as P' := P * R + t Writing this as set of scalar equations gives P'x := Px * Rxx + Py * Ryx + tx P'y := Px * Rxy + Py * Ryy + ty
If we want to express the translation within the matrix product, the tx and ty must become part of R, but then we need to multiply it with another component of the point P. This is because In a matrix product the count of columns in the left matrix (the vector P in this case) and the count of rows in the right matrix (R in this case) must be equal. This additional component need to be 1, because e.g. tx * 1 == tx doesn't change anything: P'x := Px * Rxx + Py * Ryx + tx * 1 P'y := Px * Rxy + Py * Ryy + ty * 1
But now R is no longer square and P' has 2 components but P has 3. To overcome this and to allow the resulting P' to be used in a further transformation, the same extension under consideration of the mechanics of the matrix product shows us that P'x := Px * Rxx + Py * Ryx + tx * 1 P'y := Px * Rxy + Py * Ryy + ty * 1 P'w := 1 = Px * 0 + Py * 0 + 1 * 1 will do the job. Hence, using such an extension into a 3rd dimension allows us to write P'h := Ph * Mh = Ph * Rh * Th This additional co-ordinate is named the homogeneous co-ordinate, and vectors or matrices using this are named homogeneous vectors or matrices, resp.
So: It is not only an additional dimension that allows to incorporate a translation as matrix product. It is further a special use of this co-ordinate. You'll see during your journey that there are more things with this homogeneous stuff like differing between point and direction vectors, normalized and unnormalized points, and its use for perspective projection.
Here are the methods for multiplication and inversion I wrote (just in case there are obviouse errors)... The multiplication function is just a standard row dot column loop, the inversion is using Gauss-Jordan Elimination (well; my understanding of it). Neither are written for speed... just trying to put together a working class.
Question: What is the purpose of the homogeneous coordinate (Pw)? Answer: The homogeneous coordinate (Pw) is always set to 1 and serves as a placeholder to enable the expression of translation within the matrix product. | 677.169 | 1 |
The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruentkites (also called trapezia in the US, trapezoids in Britain, or deltoids). The faces are symmetrically staggered.
The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual n-gonal antiprism has two actual n-gon faces.
An n-gonal trapezohedron can be decomposed into two equal n-gonal pyramids and an n-gonal antiprism.
In the case of the dual of a regular triangular antiprism the kites are rhombi, hence these trapezohedra are also zonohedra. They are called rhombohedra. They are cubes scaled in the direction of a body diagonal. Also they are the parallelepipeds with congruent rhombic faces.
Question: What does the 'n' in 'n-gonal trapezohedron' refer to? Answer: The arrangement of vertices around an axis of symmetry, not the shape of the faces | 677.169 | 1 |
Two circles of radii 9 and 17 centimetres are enclosed within a
rectangle with one side of length 50 cm. The two circles touch each
other, and each touches two adjacent sides of the rectangle. Find the
perimeter of the rectangle.
Given a 3,4,5 triangle, and inside it, inscribed, two circles of equal
radii. Both circles touch one leg and the hypotenuse, and both are
tangent to each other at one point. Find the radius of the circles.
Dr. Ian replies to a question about motivating students to learn about
special quadrilaterals with some excellent general advice on
motivation, math education, and real life application of mathematics.
Question: How many circles are described in the text? Answer: 4 | 677.169 | 1 |
Rotation Math
posted on: 21 Jun, 2012 | updated on: 10 Sep, 2012
In English, Rotation means to turn around an object around a center. When an object rotates around a centric Point then it is called as rotation. When any object rotates in three dimensional space about an imaginary line we call it as rotation. Just like this, we can give many definitions for a rotation. If we want to see the real world example of a rotation then we can take an example of earth. Earth revolves around the sun, this is also rotation. Moon revolves around the earth, it is also called rotation. Now the meaning of rotation is clear to you.
Now talk about the rotation in Math. In mathematics, when a rigid object revolves around the fixed point then it is called as rotation. It is just opposite of translation. We know that in translation the coordinates of an object changes and it moves from its origin to the destination location. When an object moves it could be either rotation or translation. When the object drags from one Position to another then it is translation and when is rotates around a fix point then it is rotation.
There are various application of rotation in math. Rotations math is also a type of transformation math. If coordinates are given then we can find the rotation from transformation.
Some instructions:
1: Point the coordinates on plane on x and y axis.
2: If we want to rotate the object 90 degree then just swap the x and y axis. And we will get new coordinates.
3: If we want to rotate 180 degree then there will be no swapping between x and y axis.
We have to change the positive axis into negative and negative to positive.
4: If we move the object 360 degree then we will be at the initial coordinates, there will be no change in any coordinate.
Topics Covered in Rotation Math
A geometrical figure is in Rotational Symmetry, when we say that after rotating a figure by a certain angle, the figure looks same as before. Let us look at the Rotation Symmetry of an Equilateral Triangle. By rotating the equilateral triangle, we observe that the triangle looks exactly similar when it is rotated by 120 degrees, so the rotational symmetry will be a...Read More
Matrix is a Set of similar type of values. In mathematics, matrix is used in various applications like engineering mathematics, trigonometry, geometry, differential equation, etc. We can convert an object in 3D matrix into ...Read More
Question: What happens to an object's coordinates when it is rotated 360 degrees? Answer: An object's coordinates remain unchanged after a 360-degree rotation.
Question: What is rotational symmetry? Answer: Rotational symmetry occurs when a figure looks the same after being rotated by a certain angle. | 677.169 | 1 |
Pages
2.19.2013
Made 4 Math: Distance Formula Project
Earlier in the year I talked about teaching slope with the method somebody later tagged "stack and subtract". This worked so well for me that I decided to use the method in my geometry class for the distance formula. We did a lot of practice and I decided I wanted to a do a little project instead of a standard test to assess this concept.
We did a mock version of finding the perimeter of a figure on a coordinate plane by using a simple version of @pamjwilson's idea mentioned on her blog here.
I even created an Excel file to check their work- just plug in the ordered pairs and it will calculate the distance of each segment and the perimeter of the polygon. (To know what I'm talking about you really should go read Pam's post!)
I created my scoring guide first.
But I decided to walk my students through the requirements one at a time before giving them the guide.
We started by creating a design with no more than 4 horizontal and no more than 4 vertical lines and at least 10 slanted lines. Make sure you have plenty of extra graph paper on hand.
I checked each student's design individually before giving them the next step, which was to label each endpoint with a capital letter and find the ordered pairs. Some students wrote the ordered pairs on their designs, some wrote on a separate sheet, and some wrote in the margins. I also checked these individually- it will save you a lot of time in the long run!
Next I told students they had to find the distance of each segment and then the perimeter of their entire figure. I offered copy paper or notebook paper but they had to decide how to organize their work.
Once finished with that I asked them to completely color their design and gave them the scoring guide so they could make sure they had completed all of the steps.
Here are some samples:
I again used my Excel spreadsheet to help me check the work but I'm not gonna lie, it took me some hours to grade 23 of these.
Two common mistakes: one was just not finding the distance for every segment. Sometimes this happened because they labeled incorrectly, didn't label at all, or just completely left things out.
Second was that some students listed their ordered pairs in alphabetical order and then found the distance AB, BC, CD, etc....except when you looked at their design, some of those endpoints didn't even make segments- they weren't even connected. This was probably due to the fact that it worked out that way in our 'mock' project we did the previous day, which meant that students took it for granted that it would always be that way.
That tells me that I still missed the mark. I thought I was doing something more valuable by asking them to apply their knowledge of distance but there was still a disconnect. Students still just understand how to apply a formula to numbers without making the connection that this is the distance of an actual thing.
Question: What was the main concept taught in the geometry class that the author wanted to assess? Answer: The distance formula. | 677.169 | 1 |
Pythagorean Theorem
This a program that can solve geometric problems based on Pythagorean Theorem. We suppose everybody is already familiar with the above Theorem. However, some of your kids may not have learned this theorem yet. So, let us list the formula here. By referring to a right angled triangle ABC, if the sides are AB, AC and AC respectively, where AC is the hypotenuse, then AB, AC and BC are connected by the formula AB2+AC2=BC2
Using the above formula, you can calculate the third side if the the length of any two sides are know. For example, if AB=4 and AC=3 then BC=5. We have designed the VB program for the user to input any two sides and then he or she is able to calculate the third side automatically. The third side BC can be found by finding the square root of AB2+AC2 . In visual basic, the syntax is
BC= Sqr(AB ^ 2 + AC ^ 2)
We can also the function Round to round the value to two decimal places using the syntax Round(BC, 2).
Question: What is the Pythagorean Theorem? Answer: The Pythagorean Theorem is a fundamental relation in geometry, stated as AB² + AC² = BC², where AB and AC are the legs of a right-angled triangle, and BC is the hypotenuse. | 677.169 | 1 |
I was droodling a bit and a given moment I drew the following construction:
It appears that the three blue intersections are collinear (red line), no matter how I draw the construction lines. If this is always true, I assume that this a know fact [otherwise I have my first theorem! :-) erm conjecture, since I can't prove it :-( ].
What's the theorem called?
TIA
Steven
Question: What does the acronym "TIA" stand for? Answer: "Thanks In Advance". | 677.169 | 1 |
Modern computers use a variety of techniques.[3] One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rationalapproximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction.[4] On simpler devices that lack hardware multipliers, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.
Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of radians (3°) can be found exactly by hand.
Consider a right triangle where the two other angles are equal, and therefore are both radians (45°). Then the length of side b and the length of side a are equal; we can choose . The values of sine, cosine and tangent of an angle of radians (45°) can then be found using the Pythagorean theorem:
.
Therefore:
,
.
To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:
,
,
.
Inverse functions
The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:
For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "arcsecond".
Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,
These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:
Law of sines
Law of cosines
Question: What are some examples of rational approximations used in trigonometric calculations? Answer: Chebyshev approximation, best uniform approximation, Padé approximation, Taylor series, and Laurent series. | 677.169 | 1 |
Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry everyday but often don't realize it. People use concepts of symmetry, including translations, rotations, reflections, and their geometric figures and patterns as part of their careers. Examples of people whose careers that incorporate these ideas are artists, craftspeople, musicians, choreographers, not to mention, mathematicians. It is important for students to grasp the concepts of geometry and symmetry as a means of exposing them to things they see everyday that aren't obviously related to mathematics but have a strong foundation in it. According to the National Council of Teachers of Mathematics grades 6-8 should be able to apply transformations and use symmetry to analyze mathematical situations. This includes predicting and describing the results of translating, reflecting and rotating (aka sliding, flipping, and turning) two-dimensional shapes. They should also be able to describe a motion or a series of motions that will show that two shapes are congruent, and identify and describe line and rotational symmetry in 2 and 3- dimensional shapes and designs.
This unit, "Geometry and the real world" is designed for sixth and seventh grade mathematics classes. The unit could be used in eighth grade classes as well. It will be taught over approximately 2 weeks for 90 minutes each day. The unit will cover basic concepts of geometry beginning with the core assumptions about points, lines, and planes. These are the undefined terms that will provide a starting place for basic mathematical applications used in the real world. We will also examine geometry that exists around us in the real world, both the obvious and not so obvious. Geometry deals with extensive visual reasoning and the ability to picture how certain shapes will look after being transformed into different shapes. This unit will not only bridge the gap but will also help them see how these ideas can be easily related to the environment in which they live. Once they have gotten the basic understanding of all the geometric shapes, the students will investigate isometries. The four basic isometries they will look at are; translation, rotation, reflection, and glide reflections.
All geometric diagrams are comprised of the same basic components: points, lines (and rays or line segments), planar regions. Man-made objects that are made of these geometric structures would be almost everything. If a person looks closely, they would see many geometric shapes in structures. Buildings, cars, airplanes, ships, textbooks, television sets, dishes, pictures, computers, cups all have geometric structures to name a few. Some of these (dishes and cups) are curved rather than being made of flat pieces. However, the curved ones often exhibit circular symmetry. But keep in mind that not only man-made objects are geometric. Nature has its own geometric structures. The world is a big sphere, so is the moon and the other 8 planets in the solar system. The entire world can be thought of as a geometric structure. Measurements on maps are geometric which proves that nature has geometry and that geometry exists even in things humans cannot see but we just know it's there.
Question: What are some examples of careers that incorporate the concepts of symmetry? Answer: Artists, craftspeople, musicians, choreographers, and mathematicians.
Question: What should students in grades 6-8 be able to do with transformations and symmetry according to the National Council of Teachers of Mathematics? Answer: They should be able to apply transformations and use symmetry to analyze mathematical situations, predict and describe results of translations, reflections, and rotations, describe motions to show congruency, and identify and describe line and rotational symmetry in 2D and 3D shapes. | 677.169 | 1 |
How to Find Height of Tree If Elevation and Distance is Given?
We can calculate the height of Tree mathematically if we have information about distance of tree from itself and angle of elevation. If we have these two factors then we can measure height of tree. Let us see how to find height of tree if elevation and distance is given.
Step 1: First we need to draw a triangle. This triangle must have a Right Angle and two acute angles. This triangle is formed by three sides. Hypotenuse is just opposite to angle of 90 degree.
Step 2: Now we will walk some defined distance from the tree. So this distance is known value to us. We know that how far away we are from tree. The angle between ground and tree is a right angle which is of 90 degrees. The distance is also an adjacent side of right angle.
In this triangle hypotenuse is the side or leg which is the distance from a person to top of the tree.
Step 3: Now we know that angle of elevation so it will be not be difficult to calculate the height of tree
Step 4: Elevation angle is the Ratio of ground distance and hypotenuse. We do not know the hypotenuse but we can measure it by cosine function.
Step 5: Sin function of angle is the ratio of height and hypotenuse. We have both values sin angle and hypotenuse. So now we can calculate the opposite side that is the height of tree. When we multiply the hypotenuse by value of sin angle of elevation then we will get height of the tree.
Question: What is the angle between the ground and the tree called? Answer: Angle of elevation. | 677.169 | 1 |
Distance Formula
Dr. Eaton ends this section on Radical Expressions with the Distance Formula. After studying the similarity to the Pythagorean Theorem, you will learn how to use the formula in finding missing coordinates. Four extra comprehensive examples make sure you can apply your new found knowledge.
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Distance Formula
The Distance
formula asserts that the distance between the points (x1,
y1) and (x2, y2) in the coordinate
plane is √(x2 – x1)2 + (y2
– y1)2. This formula is a special case of
the Pythagorean theorem.
Make sure that you
know and understand how to use this formula. It will be used in a
lot of later work.
In the Distance
formula, the order of the x coordinates does not matter. So take
whichever difference is easier to compute. The same comment applies
to the y coordinates.
After squaring the
differences in the distance formula, be sure to take the positive
square root of their sum.
In some problems,
you are given the distance between two points and three of their
coordinates. In this situation, use the Distance formula to find
the fourth coordinate.
Distance Formula
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Question: What should you do after squaring the differences in the distance formula? Answer: After squaring the differences, you should take the positive square root of their sum.
Question: What is the formula to calculate the distance between two points (x1, y1) and (x2, y2) in the coordinate plane? Answer: The distance formula is √((x2 - x1)² + (y2 - y1)²). | 677.169 | 1 |
Here is the triangle:
A median is a line segment drawn from the vertex of a triangle to
the midpoint of the opposite side. Since there are three vertices
of a triangle, and three sides, there are THREE medians, not just one.
Here is the median from J to KL:
Here is the median from K to JL
Here is the median from L to JK.
Now, post your problem again, this time be sure to specify exactly
what you want to find. Do you want to find the length of the three
medians? the length of just one of them? the endpoints of the medians?
Maybe you want to know where all three medians intersect, the centroid.
They all three intersect in one point, called the centroid, or center of
gravity. If you mad the triangle out of heavy cardboard, then the centroid is
the point at which the triangle would just balance if you balanced it on the
tip of a pencil.
To find the centroid we just average up the coordinates of the triangle,
(,) = (,)
Post again, telling us exactly what you are to find.
Edwin
Question: What is the method to find the centroid of a triangle given its vertices? Answer: Averaging the coordinates of the triangle's vertices | 677.169 | 1 |
to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem
Thomson problem
The Thomson problem is to determine the minimum energy configuration of N electrons on the surface of a sphere that repel each other with a force given by Coulomb's law. The physicist J. J...
). However, a useful approximation results from dividing the sphere into parallel bands of equal area
Areaand placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3
STARSHINE
The STARSHINE series of three artificial satellites were student participatory missions sponsored by the United States Naval Research Laboratory ....
.
Dividing a linecan be divided according to the golden ratio with the following geometric construction:
First, construct a line segment BC, perpendicular to the original line segment AB, passing through its endpoint B, and half the length of AB. Draw the hypotenuse AC.
Draw a circle with center C and radius B. It intsersects the hypotenuse AC at point D.
Draw a circle with center A and radius D. It intersects the original line segment AB at point S. This point divides the original segment AB in the golden ratio.
Golden triangle, pentagon and pentagram
Golden triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
CXB which is a similar triangle to the original.
If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.
Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ2. Thus φ2 = φ+1, confirming that φ is indeed the golden ratio.
Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse
Inverse (mathematics)
Question: What is the measure of angle α in the golden triangle? Answer: 36°
Question: What is the ratio of the area of the larger triangle AXC to the smaller CXB? Answer: The golden ratio (φ)
Question: What is the length of BC in the geometric construction to divide a line according to the golden ratio? Answer: BC is half the length of AB
Question: What is the length of AC in the geometric construction? Answer: AC = φ + 1 | 677.169 | 1 |
Wednesday 10/7/98
Signed (Positive and Negative) Ratios are key tools in understanding
parallels, setting up coordinates, etc.
Reading
Bix, Chapter 1, Section 1, pp. 21-38.
Assignment 3 Due
Midpoint Quadrilaterals, cubes and others.
Math 487 Lab #2 Wednesday 10/7/98
Topic: Perpendicular Bisectors, Circles and Distance
This lab will work through Chapter 3 of GTC. The main goal is to see the
connection between the perpendicular bisector of a segment and the locus of points
equidistant from the endpoints.
The main application is concurrence of
perpendicular bisectors and the constuction of the circumcircle.
Reading
GTC (Geometry Through the Circle),Chapter 3.
Friday, 10/9/98
This class will meet in the Thomson Computer Lab.
Topic: Carpenter's Construction
This lab will work through GTC Chapter 4. Key ideas are that the locus of the
of the vertex of right angles ABC for fixed A and B is a circle. This uses some
detailed geometry of the right triangle, especially the fact that the midpoint
of the hypotenuse is the circumcenter. This is applied to an introduction to more
general inscribed angles in the circle.
An important application is the construction of the tangent lines to a given circle
through a
given point exterior to the circle.
Monday, 10/12/98
Topic: Two More Concurrence Theorems
Discussion of a homework problem. Why to prove that the point
where the perpendicular bisectors of the legs interesect is the
midpoint of the hypotenuse, one needs more than equal distances, one
also needs to show the point is on the hypotenuse. This uses the
right angle in the triangle. (Otherwise, the proof turns into the
proof of perpendicular bisector concurrence for a general triangle.)
Proof of concurrence of medians of a triangle with connection to
midpoint parallelogram of a quadrilateral.
Proof of concurrence of altitudes of triangle ABC; the key is to
construct a larger triangle A'B'C' so that ABC is the midpoint
triangle of the larger triangle. The altitudes of ABC are the
perpendicular bisectors of A'B'C'. The sides of the big triangle are
parallel to the sides of ABC and distances are determined by finding
parallelograms in the figure.
Wednesday 10/14/98
Topic: Line Symmetry and Reflection with a Mirror
Using the Reflectview Mirror to reflect objects and investigate symmetry
-- an introduction. Some points include how to construct a perpendicular
bisector with this mirror. What are the lines of symmetry of familiar
figures such as an equilateral triangle (3), a rhombus (2), a rectangle (2),
a general paralellogram (0), a circle (infinitely many), a line segment (2),
Question: Which chapter of GTC is being read for the lab on Friday 10/9/98? Answer: Chapter 4
Question: What is the topic of the class on Wednesday 10/14/98? Answer: Line Symmetry and Reflection with a Mirror
Question: What is the main application of the perpendicular bisectors discussed in the lab on Wednesday 10/7/98? Answer: Concurrence of perpendicular bisectors and the construction of the circumcircle | 677.169 | 1 |
Secant right-hand side
In geometry, the relative position of two lines, or a line and a Curve, can be qualified by the adjective secant . This one comes from the Latin secare , which means to cross. In mathematical terms, a line is secant on another line, or more generally with a curve, when it has a nonempty intersection with this one.
To carry out the study of a curve in the vicinity of one of its points P , it is useful to consider the secants resulting from P , i.e. the lines passing by P and another point Q of the curve. It is starting from these secants that the concept is defined of
tangent with the curve at the point P : it is about secant the line limit, when it exists, resulting from P when the second point Q approaches P along the curve.
So when Q is sufficiently close to P , the secant can be regarded as an approximation of the tangent.
In the particular case of the curve representative of a numerical function y=f (X) , the Pente of the tangent is the limiting of the slope of the secants, which gives a geometrical interpretation of the Dérivabilité of a function.
Bond between the concepts of secant function and secant line
Let us consider a reality θ. Let us draw a secant line with the Cercle unit (centered with the origin) which passes by the origin and the point (cos θ, sin θ), not of the circle whose vector image forms an angle θ with the directing vector of the x-axis. The absolute Value of the secant trigonometrical of θ is equal to the length of the segment secant line energy of the origin until the line of equation X = 1. If the segment passes by the point (cos θ, sin θ), then the trigonometrical secant of θ is positive, if it passes by the antipodean Point, then the secant of θ is negative.
Approximation by a secant
Let us consider the curve of equation there = F ( X ) in a Cartesian Frame of reference, and consider a point P of coordinated ( C , F ( C )), and another point
Q of coordinates ( C + Δ X , F ( C + Δ X )). Then the Slope m secant line, not passing P and Q , is given by:
The member of right-hand side of the preceding equation is the report/ratio of Newton in C (or rate of increase). When Δ X approaches zero, this report/ratio approaches the derived number F ( C ), by supposing the existence of the derivative.
Question: What is the meaning of the term'secant' in geometry? Answer: In geometry,'secant' refers to the relative position of two lines or a line and a curve when they have a nonempty intersection. | 677.169 | 1 |
Fifth graders produce a curved stitching of their choice. In this symmetry lesson, 5th graders use a compass to construct a circle and bisect angles, and a protractor to measure degree intervals. Students then create a curved stitching using a needle and thread.
A 14-page packet on biseting angles, bisecting segments, constructing a perpendicular bisector, and constructing a parallel line through a point is here for you. Middle and high schoolers get lots of coaching, and lots of practice constructing these geometric elements. This unit could be used for independent work, homework, or as an assessment too. Very good!
In this geometry worksheet, students use protractors and rulers to draw 6 lines, rays, and triangles using the given measurements. They follow directions as to which lines bisects angles and other lines.
Apply geometric properties and formulae for surface area and volume by constructing a three-dimensional model of a city. Learners use similar and congruent figures and transformations to create a city of at least 10 buildings. They trade with classmates, who calculate surface area and volume of some of the structures. Refers to a Discovery Education video/DVD to support and enrich the project; a link takes you to a website to order it. Charts mentioned are not attached.
In this angle constructions worksheet, students use a straightedge and compass to construct copies of angles, bisect angles, and create other designs. This one-page worksheet contains 14 multi-step problems.
In this constructions and loci activity, students create a scale drawing of a running track. They identify the starting positions for a given race. Students bisect angles. They construct an angle trisector. This three page activity contains three multi-step problems.
Question: What is the main purpose of the angle constructions worksheet? Answer: To use a straightedge and compass to construct copies of angles, bisect angles, and create other designs | 677.169 | 1 |
Your questions at the end really get to the heart of mathematical exploration. Once we settle on one (a few?) measures of equilateralness, then the real fun begins: we can start exploring the consequences of that measure by asking question like "What kinds of triangles have the same equilateralness?" Or, "Are there some triangles that create a conflict between our intuition and our definition?"
Lovely question. It's a pretty abstract thing you're trying to measure, of course. But it's an activity that cuts to the heart of what it means to measure abstractly.
How will we know when we've got the measure right? Is it intuitively clear which one is more equilateral than the other?
Hard to sort out with this example. What about a 1-1-2 vs. a 2-2-1? Is it more transparent there? Perhaps a Twitter poll is in order to see whether you're trying to measure something that people can even perceive?
I took on a similar challenge with some preservice secondary teachers a few years back. We tried to measure the cubeyness of various prisms. This sort of thing can lead to some interesting mathematical places.
In all seriousness, it's worth exploring what happens with our proposed metrics on these extreme cases. For example, Heron's formula works perfectly well on the 1-1-2 'triangle'; maybe we shouldn't be so quick to discount it.
I retract my degenerate-triangle example and suggest 2-2-3 v. 3-3-2 instead. And again I ask the question that was the point of the example. Absent an "equilateral" measure, can we intuitively agree on which one is more equilateral? I wanna say the 3-3-2 is, but it has nothing to do with circumcenters; it's about angles. 3-3-2 is acute, where 2-2-3 is obtuse (quickly making GeoGebra sketch before posting to make sure mental image matches reality).
[...] a comment Go to comments A surprising amount of interest was generated by my question "Which Triangle is More Equilateral?" With the passing of consecutive Isosceles Triangle days, I wondered: which triangle was [...]
Among triangles, an equilateral triangle uniquely minimizes perimeter for a given area. I think that from this perspective the ratio A/P^2 is a good thing to look at. It is dimensionless, maximized for the equilateral triangle, and can be generalized to polygons with more sides.
[...] and determine which is more equilateral than the other? The post introducing the investigation is here. I encourage you to do your own exploring before reading the 28 comments which are rich in ideas. [...]
The "How square are these rectangles?" problem you've shared is a great lead-in to this problem. All the interesting questions are still there–what does 'squareness' mean? How can we measure it? Which measure should we choose? Why? Which measures are equivalent?–and the subsequent math is a bit easier to handle.
Question: What is unique about equilateral triangles in terms of perimeter and area? Answer: They uniquely minimize perimeter for a given area.
Question: What is the main point of the text regarding measuring abstract concepts? Answer: It's worth exploring what happens with proposed metrics to see if they work and are intuitive.
Question: What is one way to measure the equilateralness of a triangle? Answer: The ratio of its area to the square of its perimeter (A/P^2).
Question: Which triangle is more equilateral, 2-2-3 or 3-3-2? Answer: 3-3-2 is more equilateral. | 677.169 | 1 |
The dots are filled in because there are square brackets on each end of the interval meaning the endpoints are included. Contrast this with parentheses on the end of the interval which excludes the end point, and the graph would be an open circle rather than a filled in dot.
John
Triangles/232760: A triangle has a perimeter that is 47 cm. If side A is 3 less than twice the length of side B and side C is twice side A, what is the length of each side? 1 solutions Answer 171909 by solver91311(17077) on 2009-10-28 23:06:05 (Show Source):
Nothing here to solve or simplify. What is it you are supposed to do with these perfect square monomial terms?
John
Inequalities/232683: The rope Roz brought with her camping gear is 54 inches long. Roz needs to
cut shorter pieces of rope that are each 18 inches long. What are the
possible number pieces Roz can cut?? 1 solutions Answer 171890 by solver91311(17077) on 2009-10-28 20:16:34 (Show Source):
percentage/232616: My teacher in advanced algebra doesn't really have time to answer my questions because i have her for my 7th period. My question is,
In a catalog, a jacket was marked down from $75 to $48. What is the percent of change? Is this an increase or a decrease? 1 solutions Answer 171856 by solver91311(17077) on 2009-10-28 18:29:32 (Show Source):
Your teacher in advanced algebra has a fiduciary responsibility to answer your questions. She receives a salary to teach you, and teaching you includes answering your questions -- no matter how long it takes. If she really is telling you that she doesn't have time to answer your questions, then she is defrauding you, your parents who pay property taxes to fund her salary, and the school system that employs her. But it is your responsibility to call her on it. After all, you are the one who is not getting the education for which you are paying.
Or did you just make up this stuff about her not having time just to try to improve your chances of getting a homework problem done for you? If the real story is that you can't be bothered to stick around after 7th period to listen to the answers, then you aren't going to get much sympathy here.
The formula for percent change is: where and are the first and second given values respectively. Plug in the numbers and do the arithmetic.
Question: What is the responsibility of a teacher towards their students? Answer: A teacher has a responsibility to answer their students' questions, as they are paid to teach and educate them.
Question: In the triangle problem, what is the relationship between side A and side B? Answer: Side A is 3 less than twice the length of side B.
Question: What is the formula for calculating the percent change between two values? Answer: The formula for percent change is: ( ( - ) / ) * 100%, where and are the first and second given values respectively.
Question: What does the presence of square brackets on an interval indicate? Answer: The presence of square brackets on each end of an interval indicates that the endpoints are included. | 677.169 | 1 |
Querying
Certain transformations, such as shearing and scaling, can cause
Ellipses to become non-elliptical.
bool is_quadratic(void)
Inline const function
Returns true, because the equation
for an ellipse in the x-y plane with its center at the
origin is the quadratic equation
x^2/a^2 + y^2/b^2 = 1
where a is half the horizontal axis
and b is half the vertical axis.
Question: Which of the following transformations can cause an ellipse to become non-elliptical? A) Rotation B) Reflection C) Shearing D) Translation Answer: C) Shearing | 677.169 | 1 |
eight faces. In its most familiar form as one of the Platonic solids,
each face is an equilateral triangle.
An octahedral
pyramid has a seven-sided heptagon as its base. To get an integer
heptagon, adjacent vertices of the heptagon must be lie on a circle and
be separated by the following distances: 10, 16, 16, 10, 16, 16, and 16.
In this case, all of the points lying along a line through this circleís
center and perpendicular to the plane of the heptagon are equidistant
from the polygonís vertices. You can then choose the lateral edges of
the pyramid to be 17.
You can readily
extend the same approach to other polyhedra, such as prisms and
antiprisms. An antiprism consists of two identical polygons in parallel
planes joined in such a way that all the other faces are isosceles
triangles.
Peterson and
Jordan go on to investigate interesting links between integer octahedra
and integer antiprisms. Thereís no end to problems having to do with
integers!
Question: What are the faces of an antiprism? Answer: Two identical polygons in parallel planes joined by isosceles triangles | 677.169 | 1 |
[Test.3]Euclid's elegant proof of the Pythagorean theorem depends upon the
diagram of Figure below, sometimes referred to as the Fanciscan's cowl
of as the bride's chair. A precis of the proof runs as ? (AC)2
= 2¡âJAB = 2¡âCAD = ADKL. Similarly, (BC)2 and so on.
Fill in the detail of this proof.
Question: What is the name of the diagram used in Euclid's proof of the Pythagorean theorem? Answer: The diagram is sometimes referred to as the Franciscan's cowl or the bride's chair. | 677.169 | 1 |
Question 206886: I Found 2 solutions by jim_thompson5910, Edwin McCravy:Answer by jim_thompson5910(28598) (Show Source):
You can put this solution on YOUR website! If ABC is similar to DEF, then the corresponding sides form a ratio. In other words, the ratio of AB to DE is the same as the ratio of BC to EF. Similarly, the ratio of BC to EF is equal to the ratio of AC to DF.
What this means is and if you plug in the given lengths, you get (which is indeed true).
Furthermore, this means that and plugging in the given values gets us . So all you need to do from here is solve for 'n'
Note: because all the ratios are equal, we can also solve the equation and get the same answer.
You can put this solution on YOUR website! Edwin's solution:
I
You're supposed to do this by proportions:
(DF is to AC) as (DE is to AB) as (EF is to BC)
Write the proportion as an equation:
Substitute their lengths:
Reducing the fractions:
Cross multiply:
Edwin
Question: What is the reduced form of the equation after substituting the lengths? Answer: 3n/4 = 4n/3 | 677.169 | 1 |
You will use the distance formula to find the distance from A to B, from B to C and from A to C.
To learn about the distance formula, copy and paste the free video clip below.
If you do not understand what to do after watching the math video clip, write back.
You can put this solution on YOUR website! Let a = original number
Let b = the other digit
a + b = 6
b + a = a - 18
You have two equations in two unknowns.
Can you continue?
Trigonometry-basics/311224: The truck of a leaning tree makes an angle of 12 degrees with the vertical. To prevent the tree from falling over, a 35.0 m rope is attached to the top of the tree and is pegged into level ground some distance away. If the tree is 20.0m from its base to its top, calculate the angle the rope makes with the ground to the nearest degree. 1 solutions Answer 222571 by nyc_function(2733) on 2010-06-04 00:16:43 (Show Source):
You can put this solution on YOUR website! Di you try drawing this stuff?
Try sinx = 20.0/35
Enter 20.0 into calculator and then divide by 35. After that, press the sine inverse key to find your angle represented here by the letter x.
Can you continue?
Question: What is the formula used to find the angle the rope makes with the ground? Answer: The inverse sine function is used, i.e., sin^-1(opp/hyp) = sin^-1(20.0/35). | 677.169 | 1 |
Clinometer
Use this clinometer to determine the heights of trees and buildings or the depths of valleys. Students point the clinometer at the top of a tree or building, pull the trigger, wait for the graduated disc to stop spinning and then read the angle on the protractor. With this information, students can then calculated height by using the formula found in the guide included with the clinometer. Made of durable plastic. Grades 3-12.
Question: Who can use this clinometer? Answer: Students from grades 3-12. | 677.169 | 1 |
Easter assizes
.
The autumn assizes are regulated by acts of 1876 and 1877 (Winter Assizes Acts 1876 and 1877), and orders of council made under the former act
.
They are held for the whole of England and Wales, but for the purpose of these assizes the work is to a large extent " grouped," so that not every
equation of any degree .3 To prove the same proposition regarding it is to prove that a Euclidean construction for circle-quadrature is impossible
.
For in such a constriction every point of the figure is obtained by the intersection of two straight lines, a straight
geometry, the equations to be solved can only be of the first or second degree, it follows that the equation to which we must be finally led is a rational equation of even degree
.
Hermite' did not succeed in his
Question: Are the autumn assizes held for the whole of England and Wales? Answer: Yes | 677.169 | 1 |
In this lesson, we'll talk about similar triangles and how do we determine missing sides and angles when we are given two similar triangles. We're given two triangles PQR which is this one here and triangle XYZ and we are told that they are similar. So, triangle PQR is similar to triangle XYZ. This is the sign for similarity by the way. I'm actually is going to erase that because we don't want to confuse the sides. This is the sign of similarity. Using the given side and angle measurements that are given on the diagram, find this two.
What do we have to do? We have to find the length of the side PQ that's not given to us and the angle PQR. This is what we need to determine. We are told that these two triangles are similar. Here's what we know. For this triangle, what do we know? We know that PQ is not given to us. QR which is this side is 22 inches and PR which is the third side is 23 inches. We're also given on this triangle which is XYZ; XY is given to us as 38 inches, YZ which is this one is given to us as 44 inches and XZ is given to us just 46 inches.
We're also given all three angles here and one angle here. So, this is what we've been told. We need to find the missing angle and the missing line. Let's do this. What do we know about triangles that are similar? The two key things to remember, I'm going to use a different pen to highlight that was if the two triangles are similar then the ratio of their corresponding sides is proportional or equivalent. That's one fact and the second fact is the corresponding angles are equal. We'll actually use both of these but let's write the ratio first.
What do we know about the ratio of their corresponding sides? The corresponding sides are PQ and XY which means PQ/XY is the same as QR and the corresponding side to QR is XYZ which is equal to PR/XZ. So, all three corresponding sides of one triangle and of the other triangle are equal. The ratios are equal. Now that we know five out of the six anyway, let's substitute the values.
Question: What is the length of side PR in triangle PQR? Answer: 23 inches
Question: What is the ratio of corresponding sides PQ to XY? Answer: PQ/XY | 677.169 | 1 |
Angles of Polygons
There is a simple formula that helps determine the sum of the angles of a polygon. "Angles of Polygons" is a comprehensive geometry worksheet that helps students learn the formula, practice its use and apply the formula to solve related problems. Sixth graders will learn how to use the formula to determine the measurement of missing angles, identify regular polygons and draw conclusions about the shape of a polygon even with incomplete information about its angles.
Question: What are some of the skills students will learn from this worksheet? Answer: Students will learn to use the formula to find missing angles, identify regular polygons, and make conclusions about a polygon's shape with incomplete angle information. | 677.169 | 1 |
What does acute mean? I just wanted to know the exact meaning of acute and an example of what can be acute, ... obtuse is an angles more then 90 degrees. 2 years ago; Report Abuse; 0% 0 Votes. by Bill M Member since: June 15, 2006 Total points:
Obtuse: Acute Oblique: Basic ... (x i +1,y i +1) is given by the base times the mean height, namely (x i +1 − x i)(y i + y i +1)/2. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal.
Less than 90° - all three angles are acute and so the triangle is acute. ... Greater than 90° (obtuse): the triangle is an obtuse triangle; In the figure above, drag the vertices around and try to create all 3 possibilities.
What does Obtuse Mean? The term obtuse can be used in two different ways. An obtuse person is one who is unfeeling and insensitive to what is going on around him. An obtuse angle has a measurement of that.
What are Acute, Obtuse, Right, and Straight Angles? Note: Did you know that there are different kinds of angles? ... Do complementary angles always have something nice to say? Maybe. One thing complementary angles always do is add up to 90 degrees.
The question is If sinA=2/3 and A is obtuse, find the exact values of cosA, sin2A and tan2A. I know how to use the double angle formula and figure out cos I don't get what it means when it says A is obtuse or acute PLEASE HELP!
What Does Acute Mean? What is An Acute Triangle? What is Acute Bronchitis? What does It Mean When Your Palms Itch? What does Erratic Mean? What does Communication Mean? What does Coercion Mean? What does 86 Mean? What do IQ Test Results Mean?
What-Does-Acute-Triangle-Mean - What is the storyline of Acute Triangle ? : Add Full Plot | Add Synopsis Genres: Action ... How can you figure out if a triangle is obtuse/rightangled/acute by knowing its side lengths? If c^2 = a^2 + b^2, then it is a right triangle.
What Does Acute Mean in Medical Terms? Meaning of Acute. What Is Acute? What is the Opposite of Acute? What Is Acute Angle? What Is the Definition of Acute? Medical Dictionary. Medical Terms Definitions. What is the Definition of Acute. Instant inspiration .
Acute right and obtuse angles The difference between acute, obtuse and right angles. Practice this concept. ... it would mean that this line; is completely, if this line were the ground, this line is completely upright; relative to this line over here.
Question: What is a right angle? Answer: A right angle is one that measures exactly 90 degrees.
Question: What is the range of an obtuse angle? Answer: An obtuse angle measures between 90 and 180 degrees. | 677.169 | 1 |
between perpendicular diagonals. For the last three shapes, students will need to use a second tool, Diagonals to Quadrilaterals II. For each quadrilateral, students will describe the type (or types) of quadrilateral and explain their reasoning.
Students return to a whole-group setting. As pairs give their answers, other students are
responsible for questioning the pair for their reasoning as well as for
clarity. Probe students for responses about their conclusions. For instance, you might ask:
How do you know the quadrilateral is a rhombus?
[Students may respond that the diagonals appear to bisect each other so we can get congruent triangles like GKM and HKM using SAS.]
As students describe their findings, record their results on the Diagonals and Quadrilaterals overhead. This lists categories of quadrilaterals (some will overlap; for example, rhombi will fall in both the "general" and the "parallelograms" categories). Mark each cell with an "A" (all quadrilaterals in this category can be created given the conditions on the diagonals) or "S" (some quadrilaterals can be created).
Questions for Students
Why does it make sense that knowing the diagonals of a quadrilateral
are perpendicular is not
sufficient to show that the quadrilateral is a rhombus?
[The diagonals can be perpendicular without bisecting each
other; thus, the
quadrilateral may not be a rhombus.]
Explain using diagonals why a square is both a rhombus and a rectangle.
[A rhombus must
have diagonals that are both perpendicular and bisecting each other. A
rectangle must have diagonals that are
congruent and bisect each other. Since a square is both a rhombus and a
rectangle, its diagonals are congruent
perpendicular bisectors.]
Explain using diagonals
why a square is always a rhombus but a rhombus is not always a square.
[A square has diagonals that are congruent
perpendicular bisectors. A rhombus has
diagonals that are perpendicular bisectors. Thus, the diagonals of a
square fulfill the requirements for the
diagonals of a rhombus: perpendicular bisectors. However,
the diagonals of a rhombus need not
be congruent. So, the diagonals of a rhombus do not fulfill one of the
requirements for the diagonals of a
square: congruent.]
Assessment Options
The first objective, describe the relationships among the
diagonals and
types of figures, may be assessed in a journal
entry. A possible prompt is:
The conditions we place on the diagonals of a
quadrilateral
tell us the
type(s) of quadrilateral we have. Describe the types of conditions we
might put on
diagonals of a quadrilateral. Explain
how these conditions lead us to particular types of quadrilaterals.
An alternative way to assess this situation
Question: What is the tool used to describe the diagonals of the first three shapes? Answer: Diagonals to Quadrilaterals
Question: Which tool is used for the last three shapes? Answer: Diagonals to Quadrilaterals II | 677.169 | 1 |
SOL
Algebra ->
Algebra
-> Geometry-proofs
-> SOLLog On
Question 276811: Can you please help me!
Prove: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a paralellogram.
I started it by saying:
Given: Angle B is congruent to Angle D.
and Angle A is congruent to Angle C.
Prove: ABCD is a paralellogram.
Thank you so much!!! Answer by solver91311(16897) (Show Source):
The trick is to realize that the sum of the measures of the interior angles of any quadrilateral is 360°. So, if you have two congruent angles that each measure ° and two other congruent angles that each measure °, then the following equation must hold:
But if you divide by 2:
So now if you extend the sides of your quadrilateral past the vertices, then you can show that you have equal measure opposite interior angles formed by a transversal meaning that you have parallel lines.
Question: What shape is formed when the sides of a quadrilateral with congruent opposite angles are extended? Answer: A parallelogram. | 677.169 | 1 |
Typically, when dealing with the squares of variables, we should consider both negative and positive solutions. In a geometry problem, however, lengths must always be positive numbers, so we know that both a and c can never take on negative values. c/a is therefore simply 5/3 — it is a single value, so we therefore have sufficient data.
We can apply the same technique to statement 2). Translating the statement and rearranging, we get b=4c/5. We substitute this back into Pythagorean's theorem to find that:
Here, too, c/a = 5/3. Again, we have a single value, so we have sufficient data. Each statement alone provides sufficient data.
Our quick review of the Pythagorean theorem should have helped you get caught up on GMAT trigonometry. Before we finish, try to use what you've learned to solve the following challenge problem:
What is the ratio, by length, of AC to BC?
Right triangle ABC is divided into five identical right triangles as shown above. What is the ratio, by length, of AC to BC?
If you liked this article, let Economist GMAT Tutor know by clicking Like.
hey 45,45,90 is also a right angle triangle with two sides equal so rather than considering a,b,c as sides if i'll take sides as (AB=squareroot(2)a,BC=2AB,AC=3a) as there in case of 45,45,90 my answer will vary why so? it must be same in each case please explain?
The important relationship is that the shortest side (Green) is equal to exactly half of the second-longest side (Red). We can see this relationship in line BE. (This is why the triangles can't be 45-45-90s, as someone commented.)
Since we just need the ratio, we can use numbers. If we say Green = 1, then Red = 2. From the Pythagorean theorem, Blue = sqrt(1^2 + 2^2) = sqrt(5)
Question: Can a and c take on negative values in the given geometry problem? Answer: No, they cannot.
Question: Can each statement alone provide sufficient data? Answer: Yes, each statement alone provides sufficient data. | 677.169 | 1 |
Unit Circle
Unit Circle
Unit Circle shows the relationship between angles and trigonometric functions like sine and cosine. To do this, the program implements an interactive "unit circle" (radius = 1) diagram, where the user can click or drag to set angles and see how the values of trigonometric functions change accordingly.
The copyright and license notices on this page only apply to the
text on this page. Any software described in this text has its
own copyright notice and license, which can usually be found in
the distribution itself.
Question: What is the main purpose of the Unit Circle in the context of this text? Answer: To show the relationship between angles and trigonometric functions like sine and cosine. | 677.169 | 1 |
In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices.
This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.
This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").
You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).
Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).
This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!'ve already been using functions in algebra, but just didn't realize it. Now you will. By introducing a little more notation and a few new ideas, you'll hopefully know a function when you see one, but are curious to start looking deeper at their properties. Some functions seem to be mirror images around the y-axis while others seems to be flipped mirror images while others are neither. How can we shift and reflect them?
This tutorial addresses these questions by covering even and odd functions. It also covers how we can shift and reflect them. Enjoy!
Question: How many mini triangles of equal area are formed when the medians of a triangle intersect? Answer: Six
Question: What are the tutorials that come before and after the one on medians? Answer: Before - Perpendicular bisectors and Angle bisectors, After - Altitudes | 677.169 | 1 |
The only thing I would have certainly done differently would have been to label your answer with the given units of measurement, i.e. 13/5 cm. You might also want to consider the audience for your answer -- what would be more understandable to whoever is reading your paper: 13/5 cm or 2.6 cm? For me, I had to stop and think about how big 13/5 cm is, but just a glance at 2.6 cm tells me it is just over an inch.
John
My calculator said it, I believe it, that settles it
Triangles/577147: Please help me solve this, its for a quiz grade and im clueless on how to solve this.
Show that in a 30-60-90 triangle, the altitude to the hypotenuse divides the hypotenuse in the ratio 1:3. In triangle ABC let CD be the altitude to the hypotenuse and DB=x 1 solutions Answer 370090 by solver91311(16868) on 2012-02-23 18:14:37 (Show Source):
Presume vertex A is the 60 degree angle, then in triangle ACD, angle A is 60 degrees and angle ADC is 90 degrees, so angle ACD must be 30 degrees, therefore triangle ACD is similar to triangle ABC (by Angle-Angle-Angle). Similarly you can show that triangle BCD is also similar to ABC.
Since in a 30-60-90 right triangle the short leg is one-half of the hypotenuse, AC must be one half of AB and furthermore AD is one half of AC hence AD is one-fourth of AB. Therefore the ratio of AD to DB is 1/4 to 3/4 which is equivalent to 1:3.
There are 6 data elements, so 88 is the quotient when you divided the sum of all the data by 6. Multiply 88 by 6 to get the sum of all the data. Subtract the sum of all the known data elements. What is left will be
There are 20 grams of pure gold in 100 grams of 20% alloy. There are grams of pure gold in grams of pure gold, and we want to have 36 grams of pure gold in grams of 36% alloy.
So:
Solve for
John
My calculator said it, I believe it, that settles it
Radicals/577137: From a 220 feet high lighthouse in the Bahamas island of Nassau, a person sees a ship on the horizon. Due to the curvature of earth, how far is the ship from the lighthouse?
Use the formula: d = sqrt[3h/2], where d is the distance in nautical miles and h is the height in feet.
Question: If a lighthouse is 220 feet high, how far is the ship from the lighthouse using the given formula? Answer: First, calculate d = sqrt[3*220/2] = sqrt[330] ≈ 18.16 nautical miles. | 677.169 | 1 |
Radius and Center of a Circle from 3 Points
Date: 07/23/99 at 12:57:12
From: Nathan Sokalski
Subject: Radius and center of circle when given 3 points
I have been told during my past geometry classes that there is only
one circle that will go through any three given points. I understand
why and agree with this 100%. However, I am having trouble figuring
out what the center and radius are. Could you please tell me how to
calculate these two values?
Date: 07/24/99 at 15:51:38
From: Doctor Rob
Subject: Re: Radius and center of circle when given 3 points
Thanks for writing to Ask Dr. Math.
The equation of the circle given three points can be found in the Dr.
Math FAQ on analytic geometry formulas:
Expand the determinant, and complete the square on x and y to put it
into center-radius form. Then read off the center and radius.
Alternate approach: The center is the point equidistant from all three
points. Setting the squares of the distances equal will give you two
independent linear equations, which are the equations of the
perpendicular bisectors of two of the three line segments joining the
points. The place where those two lines intersect, that is, the common
solution of the two simultaneous linear equations, will be the center.
Then it is simple to compute the distance from the center to any of
the three points to get the radius.
- Doctor Rob, The Math Forum
Date: 07/26/99 at 01:05:48
From: Nathan Sokalski
Subject: Re: Radius and center of circle when given 3 points
When I went to the page you told me about, I did not understand what I
was supposed to do with the numbers. I assumed that the x and y (first
line) were the coordinates for the center, but what are the x, y, and
1 following the x^2+y^2 for? And I still don't understand exactly what
I am supposed to do with the numbers (maybe when I know what the x, y,
and 1 following the x^2+y^2 are for I'll figure that out, I don't
know). Thank you.
Nathan Sokalski
Date: 07/26/99 at 11:30:28
From: Doctor Rob
Subject: Re: Radius and center of circle when given 3 points
Sorry! I guess that you are not familiar with determinants. This
notation gives a compact way of writing the equation of the circle.
The less compact way gives you this equation:
(x^2+y^2-x3^2-y3^2)*(x1-x3)*(y2-y3)
+ (x1^2+y1^2-x3^2-y3^2)*(x2-x3)*(y-y3)
Question: What is an alternate approach to find the center of the circle given three points? Answer: The center is the point equidistant from all three points, which can be found by setting the squares of the distances equal and solving the resulting linear equations.
Question: Is there only one circle that can pass through any three given points? Answer: Yes.
Question: What is the compact way of writing the equation of the circle, as mentioned by Doctor Rob? Answer: Using determinants.
Question: What is the method to find the equation of the circle given three points, as mentioned by Doctor Rob? Answer: Expand the determinant and complete the square on x and y to put it into center-radius form. | 677.169 | 1 |
You can put this solution on YOUR website! What is the angle of elevation of the sun when a 35-ft mast casts a 20-ft shadow
The first thing you would do is make a triangle with the height as 35 ft and the bottom side as 20ft.
Let x= the angel of elevation
In order to find the angle of elevation, use the trigonometric ratio of tangent. tangent = opposite/adjacent
the equation will be:
tan x = 35/20
tan^-1(x) = 35/20
x = 60.2551187 degress
* I gave you the entire number including all the decimal places because you didn't specify whether or not you needed it rounded.
Hope I was of some help!
Question: What is the angle of elevation of the sun? Answer: 60.2551187 degrees | 677.169 | 1 |
Parallelograms/407605: the diagonals of a parallelogram intersect at (1,1). two vertices are located at (-6,4) and (-3,-1). find the coordinates of the other vertices. 1 solutions Answer 287337 by robertb(4012) on 2011-02-10 23:08:40 (Show Source):
Question: What is the task at hand? Answer: To find the coordinates of the other two vertices of the parallelogram. | 677.169 | 1 |
The Tangent Search Continued
Sorry, you need to install Flash and/or enable javascript in your browser to see this content. The latest version of Flash can be found at Adobe's website.
In this lesson we determine the gradient of a line through a point of interest on a curve and another point on the curve which we bring increasingly closer to the point of interest. With time we begin to observe patterns.
At the end of the lesson we introduce a spreadsheet to do the calculations that we have been doing manually throughout the lesson and show how the pattern observed earlier seems to reappear.
Question: What is the final tool introduced to perform calculations? Answer: A spreadsheet | 677.169 | 1 |
Question 177774: Analytic Geometry
On a street map, the coordinates of the two fire stations in a town are A(10, 63) and B(87, 30). A neighbour reports smoke coming from the kitchen of a house at C(41,18).
a) Which fire station is closer to this house?
b) Describe how to use geometry software to asnwer part a). Answer by Mathtut(3670) (Show Source):
Question: What is the distance from the house at C(41,18) to fire station A? Answer: The distance is 35 units. | 677.169 | 1 |
I need help with a trigonometry question plz answer!!
P, A, B and C are four points in a plane such that the angles BPA and CPA are obtuse and on opposite sides of PA. PA = 8cm, BP =10cm, PC = 12cm, AB= 14cm and AC = 18 cm. Calculate the length of BC and the area of the triangle ABC.
Question: What is the length of AC? Answer: 18 cm | 677.169 | 1 |
Parallelograms/407605: the diagonals of a parallelogram intersect at (1,1). two vertices are located at (-6,4) and (-3,-1). find the coordinates of the other vertices. 1 solutions Answer 287337 by robertb(4012) on 2011-02-10 23:08:40 (Show Source):
Question: Who provided the solution to this task? Answer: Robertb (4012). | 677.169 | 1 |
The Tangent Search Continued
Sorry, you need to install Flash and/or enable javascript in your browser to see this content. The latest version of Flash can be found at Adobe's website.
In this lesson we determine the gradient of a line through a point of interest on a curve and another point on the curve which we bring increasingly closer to the point of interest. With time we begin to observe patterns.
At the end of the lesson we introduce a spreadsheet to do the calculations that we have been doing manually throughout the lesson and show how the pattern observed earlier seems to reappear.
Question: What software needs to be installed or enabled to view the content? Answer: Flash and/or JavaScript | 677.169 | 1 |
In the figure above, points P and Q lie on the circle w/ center O. What the value of S?
The picture is an x-y axis with a semi-circle that's center is at point O (coordinates are 0,0 on the x-y axis. so the x axis is what cuts the full circle in half if you get what I mean. then there are 2 lines drawn from point O (o,o) to 2 points on the semi-circle. One drawn diagnoaly to the left, the other diagnolly to the right. These are obviously radii. The one on the left is labeled point P (-sqrt3, 1) and the one on the right is point Q (s,t). Also, the resulting angle formed from where the 2 lines are drawn from the center of the circle is a right angle.
Question: Is the angle formed by the two radii a right angle? Answer: Yes. | 677.169 | 1 |
For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two centers corresponding to the maximum and minimum sectional curvatures: these are called the focal points, and the set of all such centers forms the focal surface
Focal surface
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency...
.
For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points. There are a number of special cases. For channel surface
Channel surface
A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. One sheet of the focal surface of a channel surface will be the generating curve....In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
es and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.
All geodesics of the sphere are closed curves.
Geodesics are curves on a surface which give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.
Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
These properties define the sphere uniquely. These properties can be seen by observing soap bubble
Soap bubble
A soap bubble is a thin film of soapy water enclosing air, that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also...its surface area is minimal for that volume. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
The sphere has the smallest total mean curvature among all convex solids with a given surface area.
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.
The sphere has constant positive mean curvature.
The sphere is the only imbedded surface without boundary or singularities with constant positive mean curvature. There are other immersed surfaces with constant mean curvature. The minimal surface
Minimal surface
Question: Which solid has the greatest volume for a given surface area? Answer: The sphere.
Question: Which solid has the smallest surface area for a given volume? Answer: The sphere.
Question: What is the definition of the focal surface in the context of a surface in three dimensions? Answer: The focal surface, also known as the surface of centers or evolute, is formed by taking the centers of the curvature spheres (tangential spheres) whose radii are the reciprocals of one of the principal curvatures at the point of tangency. | 677.169 | 1 |
Isosceles Triangle Maximizes Area?
Date: 09/11/2003 at 16:31:37
From: LaShawn
Subject: isosceles triangle
How can you show that among all triangles having a specified base
and a specified perimeter, the isosceles triangle on that base has
the largest area?
I know that
b + c = p - a
where p is the perimeter. I also know that
A^2 = s(s-a)(s-b)(s-c)
and that the area is a maximum when a = b = c. But I don't know where
to go from there.
Date: 09/12/2003 at 03:58:56
From: Doctor Floor
Subject: Re: isosceles triangle
Hi, LaShawn.
Thanks for your question.
Indeed we may use Heron's formula,
A^2 = s(s-a)(s-b)(s-c)
here. Let a be the base, which is fixed, and let
2t = b + c
We may introduce the only variable x by b=t-x and c=t+x.
Substitution into Heron's formula gives:
A^2 = s(s-a)(s-t-x)(s-t+x)
= s(s-a)((s-t)^2 - x^2)
Since x is the only variable, clearly
s(s-a)((s-t)^2 - x^2)
reaches a maximum when x=0. And x=0 gives b=c=t, which makes the
triangle isosceles.
Here is a second, more geometric, way to see why the triangle will be
isosceles.
Let's fix base AB of a triangle. Then AC+BC is fixed, hence C lies on
an ellipse with foci AB. Line AB is the major axis. The points on the
ellipse at maximal distance of the major axis are on the minor axis.
These are the points such that AC=BC. By A = 1/2 bh this shows that
these points also give maximal area of ABC.
If you have more questions, just write back.
Best regards,
- Doctor Floor, The Math Forum
Question: How is 'x' introduced in the solution? Answer: As the only variable, with b=t-x and c=t+x.
Question: Is the area of an isosceles triangle always the largest among all triangles with a specified base and perimeter? Answer: Yes.
Question: What is the formula for the area of a triangle given by Doctor Floor? Answer: A^2 = s(s-a)(s-b)(s-c), where s is the semi-perimeter. | 677.169 | 1 |
UnderstandUnderstand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals
Question: What is the angle-angle criterion for similarity of triangles? Answer: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. | 677.169 | 1 |
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