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Angles/497223: I have two parallel lines with a transversal running through it. At the top (the lines are horizontal and the transversal is vertical) i have to find the measure of an angle coming of the top parallel line. The information i have to find it is that it equals 4x-30. How should i do this?? 1 solutions Answer 336692 by solver91311(16872) on 2011-09-13 20:53:32 (Show Source):
Geometry_Word_Problems/497161: A stained - glass window is being designed in the shape of a rectangle surmounted by a semicircle, as shown in the figure. The width of the window is to be 3 feet, but the height h is yet to be determined. If 4 ft2 of glass is to be used, find the height h. (Round your answer to the nearest hundredth.) 1 solutions Answer 336678 by solver91311(16872) on 2011-09-13 20:15:47 (Show Source):
If you mean that the overall area of both the rectangle and the surmounting semi-circle is supposed to be 4 ft², then this is mighty weird looking stained glass window. Since you didn't provide a dimension for the diameter of the semicircle, I have to assume that it is the same as the width of the rectangle. I'm also assuming that is the missing dimension of the rectangle since it cannot possibly be the overall height of the window. The area of a semicircle with a diameter of 3 feet and therefore a radius of 1.5 feet is . That means only of the given area remains for the rectangle. Divided by 3, that leaves a height of less than feet for the rectangle part of the window -- not even 2 inches. Standing back, you probably wouldn't notice that there was a rectanglual part of the window.
and therefore , taking the positive side because sin is positive in QII, we have
From there, the rest are easy: , , , and
John
My calculator said it, I believe it, that settles it
Percentage-and-ratio-word-problems/497088: I have tried setting this word problem into a table a few different ways, just not sure I am doing it right...
A landscaper needs to mix a 75% pesticide solution with 30 gallons of a 25% pesticide solution to obtain a 60% pesticide solution. How many gallons of the 75% solution must be used?
I got 31.5 gallons? Can that be right? Thank you! 1 solutions Answer 336649 by solver91311(16872) on 2011-09-13 19:14:56 (Show Source):
Question: How many gallons of the 75% pesticide solution should the landscaper use to achieve the desired 60% solution? Answer: 31.5 gallons
Question: What is the width of the stained glass window in the second problem? Answer: 3 feet
Question: What is the percentage of pesticide in the landscaper's initial 30-gallon solution in the third problem? Answer: 25%
Question: What is the radius of the semicircular part of the stained glass window in the second problem? Answer: 1.5 feet | 677.169 | 1 |
Euclidean & Non-Euclidean Geometries Part 2 How geometrical ideas originally were fashioned (without deductive logic), what the Greeks did to formalize geometry, and what some of our basic concepts will be. (Definitions, axioms or postulates, logic, theorems. Do they yield reality?) The two parallelograms that I drew don't use the same lengths for a and b. I also use the word equation when I really wish I had said formula. Hey, someone could take apart practically every sentence I said, but I just hope they are there when I get to the conclusion of the series to help me defend the big surprise. Because the board isn't clear, let me summarize here: EGYPTIANS: Definitions + Inspiration or experimentation = formulae which describe part of reality pertaining to interesting spacial relationships. GREEKS: Definitions + Minimal axioms + logic = Theorems Theorems + more logic = More theorems and a many more aspects of spacial relationships than can be empirically derived.
Number Theory- Greatest Common Divisor Euclidean tell me if any problems or errors as usual
Euclidean Space Lemmas Theorems Formulas Proofs PT 4-2.wmv Useful theorems proofs and concepts related to Euclidean Spaces of all dimensions. The angle between two functions from the Schwartz inequality. Functionals and Euclidean function spaces. When are orthogonal systems complete basis sets? How do you measure the dimension of a Euclidean Space? The Euclidean Isomorphism.
Euclidean Algorithm (TANTON_Mathematics) How do you find the greatest common factors of two numbers? Ask Euclid! Here we deomonstrate and explain the famous Euclidean algorithm.
Question: Which ancient civilization is associated with using definitions and inspiration or experimentation to derive formulae? Answer: The Egyptians.
Question: What is the relationship between the angle between two functions and the Schwartz inequality, according to the text? Answer: The angle between two functions is related to the Schwartz inequality. | 677.169 | 1 |
Nevertheless, if one takes the text strictly for its academic merits, One
Fold Circle is a very original and insightful look at one of geometry's most interesting subjects. While the circle (and associated folds) might not explain the nature of the universe, they certainly can help kids learn about proportions, ratios, perpendicular bisectors, and--dare I say it?--everything! (Or a lot of geometry, at any rate.)
Question: What are some of the geometric concepts that the book helps children learn? Answer: Proportions, ratios, perpendicular bisectors, and various aspects of geometry. | 677.169 | 1 |
Midpoints of Segments Lesson Packet lesson develops the concept of the midpoint of a segment and then builds slowly upon the concept so that students will be able to set up algebraic equations to find lengths of segments. Students will be engaged in hands on and minds on learning.
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Question: What is the purpose of the lesson packet, as implied from the text? Answer: To develop and build upon the concept of the midpoint of a segment gradually | 677.169 | 1 |
Three-Dimensional Figures
In this lesson our instructor talks about three-dimensional figures. First she discusses polyhedons and solids. Then she talks about prisms, platonic solids, slices and cross sections. Four extra example videos round up this lesson.
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Three-Dimensional Figures
Polyhedrons: 3-dimensional solids with all flat surfaces that enclose a single region of space
Polyhedrons have faces, edges, and vertices
Pyramid: A polyhedron that has all faces expect one intersecting at one point
Cylinder: A solid with congruent bases in a pair of parallel planes
Cone: A solid with a circular base and a vertex
Sphere: A set of points in space that are a given distance from a given point
Prism: A polyhedron with two opposite faces parallel and congruent
Platonic Solids: The five types of regular polyhedral
When a plane intersects, or slice, a solid figure, different shapes are formed
If the plane is parallel to the base of the solid, then the intersection of a cross section of the solid
Three-Dimensional Figures
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: Which of the following is NOT a type of polyhedron mentioned in the text? A) Pyramid B) Sphere C) Cube D) Cylinder Answer: B) Sphere | 677.169 | 1 |
This long equation is derived with the formulation s1+2*s2=a
and by using Pythagoras' formula twice.
......
The distance of the fixed points is c=5 and the sum a=12.
The equation is now
2304((5-x)²+y²) - (3x²+3y²-40x+44)²=0.
..............
The graph from above is incomplete. Surprisingly the equation 2304((5-x)²+y²)-(3x²+3y²-40x+44)²=0
produces another curve outside the egg curve.
...........
If you substitute m=2 with m=2.2, you produce another egg shape. You
keep c=5 and a=12.
>Trisextrix of MacLaurin y²(1+x)+0,01=x²(3-x) >Lemniskate of Bernoulli (x²+y²)²-(x²-y²)+0,01=0 >Conchoid of de Sluze 0,5(x+0,5)(x²+y²)-x²+0,02=0
...
(Torsten Sillke's idea)
Drawing by
Fritz Hügelschäffer Transfer the well known drawing of an ellipse with the help of two
concentric circles (on the left) to a two-circle-figure.
Draw in the order M1, M2
,
P1, P2 , and P.
a and b are the radii of the circles, d is the distance of its centres.
The parameters a, b, c are suitable to describe the egg shape. 2a is
its length, 2b its width and d shows the broadest position.
The equation of the egg shaped curve is
an equation of third degree:
x²/a² + y²/b²[1 + (2dx+d²)/a²] =
1
b²x²+a²y²+2dxy²+d²y²-a²b²=0
The drawn egg shaped curve has the parameters a=4, b=2 und d=1. The
equation is 4x²+16y²+2xy²+y²-64=0.
Second example:
In this example there is a=4, b=3 and d=1.
The equation is 9x²+16y²+2xy²+y²-144=0.
Origin: (11), page 67/68
Granville's Egg Curve
>There is given a line, which starts at point A and lies horizontally.
Then there is a vertical line in the distance of a and a circle with the
radius r being symmetric to the horizontal line in the distance of a+b
(drawing on the left).
>If you draw a line (red) starting at point A, it cuts the vertical
Question: Which mathematical formula was used twice to derive the long equation? Answer: Pythagoras' formula
Question: What is the sum of the distances from the fixed points in the given equation? Answer: 12
Question: Which curve does the equation 2304((5-x)²+y²) - (3x²+3y²-40x+44)²=0 produce? Answer: Another curve outside the egg curve | 677.169 | 1 |
line at B and the circle at C. If you draw then a vertical line through
C and a horizontal line through B (green), they meet at P.
>If the point C moves along the circle, then the points P lies on a
egg shaped curve (animation on the right).
See more: (13), Jan Wassenaar (Granville's egg, URL below), Torsten
Sillke (Granville's egg, URL below)
Mechanical Egg Curve
Construction
Let P be a fixed point and A a point, which moves on a circle around
P with the radius r=PA.
Link the bar a=QA at A . Its free end Q moves on a horizontal through
P forth and back. The point B on the line AQ with BQ=b describes an egg
shaped curve.
Two small (red) and two big (grey) quarter circles, which have a square
in common, form an oval.
(The angles of the sectors don't have to be 90°.)
......
A semi-circle (green), a quarter circle (red) and two eighth circles
(grey), which have a triangle in common, form a second figure. If you cut
the egg in nine pieces, you get the tangram puzzle "The Magic Egg" or "The
Egg of Columbus".
......
You can generalize the figure: Take a smaller dark grey triangle.
...
Divided and reassembled again
......
Divided and reassembled again.
(14), Seite 122..
Section through Rotation
Shapestop If you make a sloping section through a cone
or a cylinder you often get an ellipse as a section line. If you choose
an hyperbolic funnel, you get egg curves in the form of a hen egg. Hyperbolic
funnels are figures, which develop from rotation of an hyperbola around
the symmetry axis.
...
There is the hyperbolic funnel to f(x) =1/x². The y-axis is perpendicular to the x-z-plane
in direction to the back.
The straight line shows the section plane perpendicular to the x-z-plane.
...
The given plane intersects the hyperbolic funnel
with
three points in the x-z-plane.
If you project the section lines in the x-y-plane you get the red curves.
...
You get an egg curve in the section plane.
Formulas:
If you make a sloping section through other figures,
you get more egg curves.
Question: Who are the sources mentioned for more information about Granville's egg? Answer: Jan Wassenaar and Torsten Sillke.
Question: What happens if you make a sloping section through other figures? Answer: You get more egg curves.
Question: What is the result of projecting the section lines in the x-y-plane? Answer: The red curves. | 677.169 | 1 |
Parabolas Related Terms
Parabolas Word Problems
Submit your word problem:
Grade Level: Subject:
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Eighth Grade: Parabolas Word Problems
A cable of a suspension bridge is in form of a parabola
A cable of a suspension bridge is in form of a parabola whose span is 40 m. the roadway is 5 m below the lowest point of cable. If a extra support is provided across the cable 30 m above the ground level, find length of the support if the height of the pillars are 55 m
Determine the distance across the base
Determine the distance across the base of a parabolic arch that measures 30 feet at its highest point if the roadway through the center of the arch is 48 feet wide and must have a minimum clearance of 15 feet.
A mirror in the shape of a paraboloid
A mirror in the shape of a paraboloid of revolution. If a light source is located 1 foot from the base along the axis of symmetry and the opening is 2 feet ...
Question: What is the shape of the mirror described in the third word problem? Answer: A paraboloid of revolution. | 677.169 | 1 |
Question 3
Prove that the triangle above exists.
[25 marks]
Question 4
What is the area of a octagon of side length y, in cubic inches. (Note that this question uses non-euclidean goemetry)
[2πr marks]
Question 5
Through cunning use of Pythaogoras' Theorem, prove that aliens do not exist.
[-0 marks]
Question 6
If Failed to parse (lexing error): 48y^2+πy-6778=x
, then what does y smell like?
Question: What is the value of π in the context of the text? Answer: The value of π is not given in the text, it's just mentioned in the context of the area of an octagon. | 677.169 | 1 |
With this song, as is true with most of the others, students need clear
instruction that includes manipulatives and plenty of practice. I have found
that the basic concept of, "What is an angle?" can be very confusing for
some students.
I believe that the hand motions on this song are absolutely essential. After
all, students are singing, "Square corner just like so." If students don't
show the angle with their hands, then what do they mean when they sing,
"...just like so?" Watch a few students
in my class doing the hand motions to this song.
Question: If students don't use hand motions, what does it imply about their understanding of the line "Square corner just like so"? Answer: It implies that they might not understand what they mean when they sing that line. | 677.169 | 1 |
a line, segment, or ray in the plane of a circle that intersects the circle in exactly one point
point of tangency
the one point where the tangent intersects the circle
tangent circles
coplanar circles that intersect in one point
concentric circles
coplanar circles that have a common center
common tangent
a line, ray, or segment that is tangent to two coplanar circles
Perpendicular Radius to a Tangent Theorem
In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.
Congruent Tangents to External Point Theorem
Tangent segments from a common external points are congruent.
central angle
an angle whose vertex is the center of the circle
minor arc
an arc of a circle whose measure is less than 180 degrees
major arc
an arc of a circle whose measure is more than 180 degrees
semicircle
an arc with endpoints that are the endpoints of a diameter
measure of a minor arc
measure of its central angle
measure of a major arc
360-measure of related minor arc
measure of a semicircle
180
arc addition postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
congruent circles
circles with the same radius
congruent arcs
arcs with the same measure that are part of the same circle or of congruent circles
Congruent Corresponding Chords and Minor Arcs Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Perpendicular Bisecting Chords & Arcs Theorem
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
Perpendicular Bisecting Chords & Arcs Converse
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Equidistant Congruent Chords from the Center Theorem
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
inscribed angle
an angle whose vertex is on a circle and whose sides contain chords of the circle
intercepted arc
the arc that lies in the interior of an inscribed angle and has endpoints on the angle
Measure of an Inscribed Angle Theorem
The measure of an inscribed angle is one half the measure of its intercepted arc.
Congruent Inscribed Angles with the Same Arc Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Hypotenuse of an Inscribed Right Triangle is a Diameter Theorem
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Hypotenuse of an Inscribed Right Triangle is a Diameter Converse
Question: What is the measure of a semicircle? Answer: 180 degrees.
Question: What happens to the hypotenuse of an inscribed right triangle? Answer: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. | 677.169 | 1 |
The answer is "D".
Stat(1): AC=EG. Let EG=x. then EH=x/sqrt(2). AB=x/2(sqrt2).With this we can find the side BC(one side of the rectangle and the diagonal are known).Hence we can find the area in terms of 'x'. Hence suffic.
Stat(2): As the diagonal divides both into two equal parts, if area of triangles is not same, then definitely the areas of rectangles are not same.
Hence suffic....
one thing: I did this by plugging in numbers. The square is a perfect figure, and the rectangle's side is half the square, so I knew that by plugging in just one value I'd get a very good idea of what's happening in the problem. Once I did, I saw everything clearly.
Question: What does the author do after plugging in one value? Answer: Sees everything clearly | 677.169 | 1 |
The problem is different interpretations
of three "random" points, and choosing from an infinite
interval:
(a) Put A and B on the x-axis and draw
vertical lines up from each, dividing the upper half plane into
regionsT1, T2, T3. Draw a semicircle
above x-axis with diam. AB. Then assuming (WLOG) that C is above
the x-axis,if C in T1, angle A
is obtuse; if C in T2, angle B is obtuse, and if C is in the
semicircle, angle C is obtuse.The
only region for C to be acute is in the strip T2 above the circle;
this area has finite width compared to theinfinite
width regions T1 and T3, therefore the ratio of the acute to
the total is zero; Prob(obtuse) = 100%.
(b) Let the three trees be A, B, and
C, and assume side AB is longest. Draw circles centered at A
and B with radius AB. The lune shape must contain C as AB is
longest. Now draw a circle with AB as diameter. If C isinside, on, or outside this circle then
C is an obtuse, right, or acute angle respectively. The ratio
of the areas ofthe circle to the
lune is [pi r^2] / [(8 pi - 6\/3) / 3] ~ 0.63938... which
is the prob that /_\ ABC is obtuse.
(c) Three points determine a circle,
so let's assume three random points on the circumference. Say
tree A is
at the 0 degree mark; then B and C
will be somewhere between 0 and 360, say distributed uniformly.
If B and C are both less than or both
more than 180, the triangle will be obtuse; prob 1/4 for each.
If B and Care
more than 90 apart the triangle will be obtuse, prob 1/4. Total:
3/4 = 0.75 in this scenario.
(d) The angles add to 180: A+B+C =
180 is a plane in 3-space (A,B,C); use the slanted equilateral
triangle
where all A, B, and C are >= 0.
In this region /_\ ABC will be obtuse if A > 90 or B >
90 or (180-A-B) > 90.
This makes 3/4 of the region, so the
prob ABC is obtuse is again 0.75.
(e) AB might be the long side, the
middle side, or the short side. Compute the prob of obtuse for
each and
average the results: (0.63938 + 0.82102
+ 1) / 3 = 0.82013, very close to that middle value!
Question: What is the total probability of an obtuse angle in scenario (c)? Answer: 0.75
Question: Which of the following is NOT a method used to determine the probability of an obtuse angle? A) Drawing regions on a plane B) Using a slanted equilateral triangle C) Considering the distribution of points on a circle D) Using a random number generator Answer: D) Using a random number generator
Question: What is the average probability of an obtuse angle in scenario (e)? Answer: 0.82013
Question: In scenario (c), what is the probability that the triangle will be obtuse if both B and C are more than 180? Answer: 1/4 | 677.169 | 1 |
Each pair of lines crosses at only one point (E1)
and divides the plane into two lunes with their four vertices touching at this
point, figure f/2.
Of the six lunes, we focus on the three shaded ones, which overlap the triangle.
In each of these, the two interior angles
at the vertex are the same (Euclid I.15).
The area of a lune is proportional to its interior angle, as follows from dissection into narrower lunes;
since a lune with an interior angle of π covers the entire area P of the plane, the constant
of proportionality is P/π.
The sum of the areas of the three lunes is (P/π)(α+β+γ), but these three areas also cover the entire plane,
overlapping three times on the given triangle, and therefore their sum also equals P+2A.
Equating the two expressions leads to the desired result.
This calculation was purely intrinsic, because it made no use of any model or coordinates. We can therefore
construct a measure of curvature that we can be assured is intrinsic, K=ε/A. This
is called the Gaussian curvature, and in elliptic geometry it is
constant rather than varying from point to point. In the model on a sphere of radius ρ, we have
K=1/ρ2.
Self-check: Verify the equation K=1/ρ2 by considering a triangle covering one octant of the sphere,
as in example 2.
It is useful to introduce normal or Gaussian normal coordinates,
defined as follows. Through point O, construct perpendicular geodesics, and define affine coordinates x and y along
these. For any point P off the axis, define coordinates by constructing the lines through
P that cross the axes perpendicularly. For P in a sufficiently small neighborhood of O, these lines exist
and are uniquely determined. Gaussian polar coordinates can be defined in a similar way.
Here are two useful interpretations of K.
1. The Gaussian curvature measures the failure of parallelism in the following sense. Let line ℓ
be constructed so that it crosses the normal y axis at (0,dy) at an angle that differs from
perpendicular by the infinitesimal amount dα (figure h).
Construct the line x'=dx, and let dα' be the angle its perpendicular forms with ℓ.
Then4
the Gaussian curvature at O is
where d2α=dα'-dα.
2. From a point P, emit a fan of rays at angles filling a certain range θ of angles in Gaussian polar coordinates (figure i).
Let the arc length of this fan at r be L, which may not be equal to its Euclidean value LE=rθ.
Then5
Let's now generalize beyond elliptic geometry. Consider a space modeled by a surface embedded in three dimensions, with
geodesics defined as curves of extremal length, i.e., the curves made by a piece of string stretched taut across the surface. At a particular
point P, we can always pick a coordinate system (x,y,z) such that the surface
Question: What is the relationship between the area of a lune and its interior angle? Answer: The area of a lune is proportional to its interior angle.
Question: What does the Gaussian curvature measure in terms of parallelism? Answer: It measures the failure of parallelism, i.e., the difference in angles of two lines constructed at an infinitesimal angle from perpendicular. | 677.169 | 1 |
Plato''s Impossible triangle
Plato''s Cursed Triangle (The Impossible Triangle)
The Impossible Triangle is an optical illusion. This explanation is written in plain English since the geometric equations behind this illusion are too complex to be understood by anyone but math majors. Simply put: The triangle that is not complete is not really a triangle at all, it just looks like one. It''s a fake triangle. The big fake triangle seem to form a straight diagonal line from the lower left corner to the upper right corner, like they do in the real triangle. But in reality it''s not a straight line in the fake triangle. You can compare the real triangle with the fake triangle. Look at their upper diagonal sides and you will see that the way they cross the white grid in the background is slightly different. There is an invisible dent where the two parts or pieces meet in the fake triangle. As strange as it may seem, the slight dent is enough to throw off the straight line, and enough to amount to the volume of the missing square.
Question: What makes the Impossible Triangle 'impossible'? Answer: It appears to be a complete triangle, but it is not, as it is missing a square. | 677.169 | 1 |
Pythagoras
Pythagoras (c.580-500 BC) was a Greek mathematician and philosopher, who formulated Pythagoras's theorem: in a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
Short Biography:
Much of his work concerned numbers, to which he assigned mystical properties. For example, he classified numbers into triangular ones (1, 3, 6, 10,...), which can be represented as a triangular array, and square ones (1, 4, 9, 16,...), which form squares. He also observed that any two adjacent triangular numbers add to a square number (for example, 1 + 3 = 4, 3 + 6 = 9, 6+ 10= 16,...).
Question: Who was Pythagoras? Answer: Pythagoras was a Greek mathematician and philosopher. | 677.169 | 1 |
TRIGONOMETRY TUTORIAL
The classical concept of trigonometry deals with the relationships between the angles and sides of triangles. Over time, however, trigonometry has been adapted so that the angles do not necessarily represent angles in a triangle. For example, in calculus, trigonometric functions are defined for arbitrary real numbers.
Angles can be expressed in degrees or radians. To convert a measurement from radians to degrees (or vice versa) we use the following relationship:
This relationship gives the following two equations:
Note: By convention, most angles are expressed in radian measure, unless otherwise stated.
Note: Make sure that your angles are measured in radians. The arc length formula does not hold for angles measured in degrees. Use the conversion relationship above to convert your angles from degrees to radians.
Note: For an example of an arc length question, see question #1 in the Additional Examples section at the bottom of the page.
——————————————————————————–
Trigonometric Functions
The following ratios are for right angle trigonometry. The angle must be acute (angle is less than 90°).
For angles that are obtuse (angle is greater than 90°) or negative, we use the following trigonometric ratios. The x and y variables are the values of the x and y coordinates, respectively. The r variable represents the distance from the origin, to the point (x,y). This value can be found using the Pythagorean theorem.
When negative or obtuse angles are used in trigonometric functions, they will sometimes produce negative values. The CAST graph to the left will help you to remember the signs of trigonometric functions for different angles. The functions will be negative in all quadrants except those that indicate that the function is positive. For example, When the angle is between 0° and 90° (0 and pi/2 radians), the line r is in the A quadrant. All functions will be positive in this region. When the angle is between 90° and 180° (pi/2 and pi radians), the line is in the S quadrant. This means that only the sine function is positive. All other functions will be negative.
Note: For examples of finding trigonometric ratios see questions #2 and #3 in Additional Examples at the bottom of the page.
——————————————————————————–
Graphs of Trigonometric Functions
Sine Function
Cosine Function
Tangent Function
Note: As illustrated in the graphs above, the sine and cosine functions are defined for all values of x. The tangent function, however, being equal to sin x / cos x, is undefined whenever cos x = 0.
——————————————————————————–
Periodicity of Trigonometric Function
From the graphs above, you can see that trigonometric functions are periodic. The sine and cosine functions, for example, have a period of 2 pi. In general, for any integer k,
The tangent function, however, has a period of pi. The period of the tangent function is given by
for any integer k.
Question: What is the period of the sine and cosine functions? Answer: 2π
Question: In which quadrant(s) are all trigonometric functions positive? Answer: The A quadrant (0° to 90° or 0 to π/2 radians) | 677.169 | 1 |
Problem :
When a line is drawn through an equilateral triangle such that it is parallel to one side and
intersects one time with each of the other two sides, how many similar pairs of triangles
are created?
Thirty-six. If the equilateral triangle is triangle ABC, and the line intersects with sides
AC and BC, then a new triangle, DFC, is created. Let point D be the intersection point of
the line and side AC and let point E be the intersection point of side BC. Then triangle
DFC is similar to triangles ABC, BCA, CAB, ACB, BAC, and CBA. The five other
triangles, FCD, CDF, DCF, FDC, and CFD are also all similar to these six triangles,
making 36 pairs of similar triangles.
Question: How many additional triangles are formed by the line intersecting the sides of the equilateral triangle? Answer: 5 | 677.169 | 1 |
An isosceles triangle is a triangle with (at least) two equal sides, then base angles are congruent.
let the two equal angles be and
so,
third angle is
if the third angle of an isosceles triangle is dgree less than the sum of the two equal angles, then
......since , we have
..solve for
we know that the sum of all three angles in triangle is
then we have:
.......substitute and ....solve for
then
and
hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (2, -1) and (x2, y2) = (3, 4).
Slope a is .
Intercept is found from equation , or . From that,
intercept b is , or .
Question: Which of the following is NOT a property of an isosceles triangle? A) All sides are equal, B) The base angles are equal, C) The vertex angles are equal. Answer: A) All sides are equal. | 677.169 | 1 |
Overview: The Spidron is a planar figure consisting of two alternating sequences of isosceles triangles which, once it is folded along the edges, exhibits extraordinary spatial properties. The Spidron can be used to construct various space-filling polyhedra and reliefs, while its deformations render it suitable for the construction of finely adjustable dynamic structures.
With details: A spidron is a plane figure consisting of an alternating sequence of equilateral and isosceles (30°, 30°, 120°) triangles. Within the figure, one side of a regular triangle coincides with one of the sides of an isosceles triangle, while another side coincides with the hypotenuse of another, smaller isosceles triangle. The sequence can be repeated any number of times in the direction of the smaller and smaller triangles, and the entire figure is centrally projected through the mid-point of the base of the largest isosceles triangle.
Question: Which side of a regular triangle coincides with the hypotenuse of another isosceles triangle? Answer: Another side of the regular triangle. | 677.169 | 1 |
HP 50g Calculator - Changing the Angle Measure Mode
The calculator has three setting options for the angular mode for use with trigonometric functions:
Degrees: A circle is divided into 360 degrees (360°).
Radians: The circumference of a circle is 2π radians.
Grades: There are 400 gradians (400 g) in a circle.
The angular mode affects functions like sine (SIN), cosine (COS), tangent (TAN) and other associated trigonometric functions. To change the angular mode, use the following procedure:
Press the
button.
Use the down arrow key,
, twice.
To select the desired mode, either:
Cycle through the options by pressing the
key. The displayed option is the active setting.
Press the
soft menu key to open a Choose box.
NOTE:
To select an item in the menu along the bottom of the display screen, press the key directly below the displayed item.
To select
, for example, press the 'F2' key. Use the up and down arrow keys,
, to highlight the preferred angular mode, and press the
soft menu key to select it. The displayed option is the active setting. In the screen shown below, the 'Radians' mode is selected.
Figure 1: Selecting the Radians mode
Press the
soft menu key to save the setting and exit the 'CALCULATOR MODES' page
Question: What are the three angular modes available on the HP 50g calculator? Answer: Degrees, Radians, and Grades. | 677.169 | 1 |
G.SRT.1.a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G.SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.2 GivenG.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G.SRT.4-G.SRT.5 Prove Theorems Involving Similarity
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures that
Question: True or False: A dilation can change the length of a line segment that passes through the center of dilation. Answer: False. | 677.169 | 1 |
Seven Sided Polygon
Embed or link this publication
Description
A polygon is a geometrical shape which is constructed by the straight lines. In geometry there
are various kinds of polygons are defined that are constructed with help of straight lines.
Polygon is a special kind of geometrical shape that has equal no.
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p. 1
seven sided polygon seven sided polygon a polygon is a geometrical shape which is constructed by the straight lines in geometry there are various kinds of polygons are defined that are constructed with help of straight lines polygon is a special kind of geometrical shape that has equal no of line segment and same amount of connecting points in mathematical definition we can describe the polygon as a geometrical shape which is enclosed with the several line segments in the polygons the connecting points are popularly called as vertex point and in the same aspect the line which connects the two vertex point to each other are known as edges of the polygon in the polygon there is no limit for no of line segment that are exists into the polygon in the formation of polygon it is necessary that to form the polygon with three sides.according to above definition of polygon we can say that triangle is a geometrical shape which can be consider as a polygon because triangle is a shape which is formed by three straight lines normally in the concept of polygon we discuss about the regular polygon regular polygon is a type of polygon in which all the sides and all the angles are equal in the measure know more about how to divide decimals math.tutorvista.com page no 1/4
p. 2
Question: What does the term 'edges' refer to in a polygon? Answer: The lines that connect two vertex points in a polygon.
Question: What is a triangle classified as in the context of polygons? Answer: A triangle is considered a polygon. | 677.169 | 1 |
Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the volume of the tetrahedron defined by them is 0,
Coplanarity is equivalent to the statement that the pair of lines determined by the four points are not skew, and can be equivalently stated in vector form as
An arbitrary number of points , ..., can be tested for coplanarity by finding the point-plane distances of the points , ..., from the plane determined by and checking if they are all zero. If so, the points are all coplanar.
A set of vectors is coplanar if the nullity of the linear mapping defined by has dimension 1, the matrix rank of (or equivalently, the number of its singular values) is (Abbott 2004).
Parallel lines in three-dimensional space are coplanar, but skew lines are not !!!!
Question: What is the condition for four points to be coplanar? Answer: The volume of the tetrahedron defined by them is 0 | 677.169 | 1 |
List the values of sin(α),
cos(α), sin(β), and tan(β)
for the triangle below, accurate to three decimal places:
For either angle, the hypotenuse has length 9.7. For the angle α, "opposite" is 6.5 and "adjacent" is 7.2, so the sine of α will be 6.5/9.7 = 0.6701030928... and the cosine of α will be 7.2/9.7 = 0.7422680412....
For the angle β,
"opposite" is 7.2 and "adjacent" is 6.5, so the sine of β will be 7.2/9.7 = 0.7422680412... and the tangent of β will be 7.2/6.5 = 1.107692308.... Rounding to three decimal places, I get:
sin(α) = 0.670,
cos(α) = 0.742, sin(β) = 0.742, tan(β)
= 1.108
Once you've memorized the trig ratios, you can start using them
to find other values. You'll likely need to use a calculator. If your calculator does not have
keys or menu options with "SIN", "COS", and "TAN", then now is the
time to upgrade! Make sure you know how to use the calculator, too; the owners manual should have
clear instructions.
In the triangle shown below, find the value
of x, accurate to three decimal places.
They've given me an angle measure and the length of the side "opposite"
this angle, and have asked me for the length of the hypotenuse. The sine ratio is "opposite
over hypotenuse", so I can turn what they've given me into an equation:
I have to plug this into my calculator to get the value of x: x = 190.047286...
x = 190.047
Note: If your calculator displayed a value of 71.19813587..., then check the "mode": your
calculator is set to "radians" rather than to "degrees". You'll learn about
radians later.
For the triangle shown, find the value
of y, accurate to four decimal places.
They've given me an angle, a value for "adjacent",
and a variable for "opposite", so I can form an equation:
tan(55.3°) = y/10 10tan(55.3°) = y
Plugging this into my calculator, I get y = 14.44183406....
y = 14.4418
Question: What is the value of x, accurate to three decimal places, for the given triangle? Answer: 190.047 | 677.169 | 1 |
degrees.
There is no easy relationship between these three angles, just like
there is no easy relationship between the distances between three
points. If you know that there is 23.4 km between cities A and B,
then you don't have enough information to say how much distance there
is between A and C, or between B and C. If you also know that there
is 60.2 km between A and C, then you still don't have enough
information to say precisely how much distance there is between B and
C: It could be as little as 60.2 - 23.4 = 36.8 km (if B is exactly
between A and C) or as much as 60.2 + 23.4 = 83.6 km (if A is exactly
between B and C), or any value in between, depending on the angles
inside triangle ABC. Something similar goes for the three celestial
planes.
where l1 and b1 are the longitude and
latitude of the first point, and l2 and
b2 the longitude and latitude of the second point
(with + or - for east or west longitude and north or south latitude).
The result d is then the (shortest) angular distance
between the two points across the sphere. If the angular distance is
measured in degrees, and if the radius of the sphere is equal to
R (for example, measured in kilometers), then the
distance D across the sphere (in the same units as
R) is equal to
(Eq. 2) D = dR * (π/180) ≈ 0.01745329 dR
For example: The distance between the Netherlands (52° north latitude,
5° east longitude) and Mexico City (19° north latitude, 99° west
longitude) is equal to d = arccos(sin(52°)*sin(19°) +
cos(52°)*cos(19°)*cos(5° − (−99°)) = 83.35°. The radius of
Earth is 6378 km, so this distance corresponds to D =
0.01745329*83.35*6378 = 9279 km.
Distances that you calculate across the Earth in this way are not
entirely correct, because the Earth is not a perfect sphere. The
error will be less than half a percent in general.
If the two points have almost the same longitudes and also almost the
same latitudes, then they are close together on the sphere. In that
case the results of equation 1 might be inaccurate because
of rounding errors. You can then use the following alternative
formula:
The point in the sky that is straight above you (or the direction that
is straight up) is called the zenith. The point that is straight
below you (or the direction that is straight down) is the nadir. The
Question: What is the error in calculating distances across the Earth using the given formula? Answer: Less than half a percent.
Question: What is the formula to calculate the angular distance between two points on a sphere? Answer: d = arccos(sin(lat1)sin(lat2) + cos(lat1)cos(lat2)*cos(lon1 - lon2)).
Question: Which points in the sky are the zenith and nadir? Answer: The zenith is the point straight above you, and the nadir is the point straight below you. | 677.169 | 1 |
Statement 1 shows the ratio between each of the sides. AB:BC:AC is 6:4:3. We don't know the lengths, by using the ratio we can find the angles. I think the only way is by using trigonometry. Sufficient.
Statement 2, if it were = instead of >, would be a right triangle, meaning that no, one of the angles is not smaller than 90 degrees. But if the lengths of sides AB and AC increase, but BC (hypotenuse) does not, angle A will get smaller, making the answer yes. I had to draw a right triangle on paper to really see this. Insufficient.
Question: Can we determine the lengths of the sides with the given ratio? Answer: No, we cannot determine the lengths of the sides with the given ratio. | 677.169 | 1 |
Unit Summary
In this unit students discover the Pythagorean Theorem and explore its implications. However, these students with Specific Learning Disabilities may lack the prior knowledge needed of angle measurement, area of triangles, rules of exponents, and estimating roots. Breaking this unit into smaller chucks allows the students the opportunity to gain a more complete understanding of the Pythagorean Theorem
21st Century Skills xCreativity and Innovation- Egyptian Knotted Ropes
Media Literacy
x Critical Thinking and Problem Solving- How to create a triangular building after viewing an informational website selected by the teacher.
x ICT Literacy: Smart-Board, document camera, Geo-Sketch Pad software
Communication and Collaboration
x Life and Career Skills: Explore real life careers that use the Pythagorean Theorem
Information Literacy: Carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers of many sorts need to know it. One time I observed people who needed to measure and mark on the ground exactly where the building would go. They had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a parallelogram. It's not easy to be sure that you have really drawn the two sides in a right angle.
Interdisciplinary Connections: World History of Ancient Greece and Ancient Egypt
Integration of Technology: Using the Smart Board with GeoSketch Pad
Equipment & materials needed: colored pencils, pencils, grid paper, dot paper, calculators. Rulers (straight edge), Connect Math text books pages 27-28. Smart-Board, document camera, Geo-Sketch Pad software, and internet access Teacher Resources:document camera, student work, colored pencils, pencils, grid paper, dot paper, calculators. Rulers (straight edge), Connect Math text books pages 27-28. Smart-Board, document camera, Geo-Sketch Pad software and internet access.
Goals/Objectives/CPIs
Students:
Work out the Pythagorean Theorem through exploration and use the Pythagorean Theorem to find areas of squares drawn on a dot grid with at least 80% accuracy.
Will be able to identify that a right triangle 2 legs and a hypotenuse with at least 80% accuracy.
Learning Activities/Instructional Strategies
Engagement:
Begin with reviewing the terms square roots and area.
Next place the text book on the document reader and point out and explain the legs and hypotenuse of a right triangle.
Now discuss 3.1 on pg.27 discovering the Pythagorean Theorem.
Introduce the idea of drawing squares on the sides of a right triangle and comparing their areas as shown on page 27 in their text books Investigation 3 the Pythagorean Theorem.
Now have students draw on dot paper 3 different size right triangles and look for the squares of their triangle and figure out the areas of each.
Question: What is the diagonal check used for in construction? Answer: To ensure that the building is really going to be a rectangle and not a parallelogram
Question: What is the purpose of breaking the unit into smaller chunks? Answer: To allow students to gain a more complete understanding of the Pythagorean Theorem
Question: What is the minimum accuracy expected from students when working out the Pythagorean Theorem? Answer: 80% | 677.169 | 1 |
geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constantCharacteristicsby the fact that successive turnings of the spiral have a constant separation distance (equal to 2π.
The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axis will yield the other arm.
General Archimedean spiral
Sometimes the term Archimedean spiral is used for the more general group of spirals
The normal Archimedean spiral occurs when x = 1. Other spirals falling into this group include the hyperbolic spiral
Hyperbolic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral
Parker spiral
The Parker spiral is the shape of the Sun's magnetic field as it extends through the solar system. Unlike the familiar shape of the field from a bar magnet, the Sun's extended field is twisted into an arithmetic spiral by the magnetohydrodynamic influence of the solar windThe Catherine wheel is a type of firework consisting of a powder-filled spiral tube, or an angled rocket mounted with a pin through its centre...
) are Archimedean.
Constructing an Archimedean spiral
This is an extended geometrical construction, where a constant rate of expansion of the compass is applied during rotation of the compass. To generate an Archimedean spiral by hand for arts crafts, one may follow the instructions given in "Handicraft For Boys" by A. Frederick Collins, 1869.
"Make a loop in one end of a thread as before and tie the other end tightly to a large pin; wind the thread around the pin until all of it is on except the loop; push the pin through the paper on which you want to draw the spiral and into the drawing board [...]
Next put the point of the pencil in the loop and move it around the pin just as you did in making the circle and you will find that you have drawn a very pretty geometrical spiral which is known as the spiral of Archimedes. It is so called because Archimedes was the first to explain that it was caused by a point moving with uniform angular speed and receding from the center at a constant rate."
Applications
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
, by relaxing the strict limitations on the use of straightedge and compass in ancient Greek geometric proofs, makes use of an Archimedean spiral.
Question: What is an Archimedean spiral? Answer: An Archimedean spiral is a curve that results from a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. | 677.169 | 1 |
Origin
The origin, of something, is the location from which it originated; that is, from whence it came.
The origin (from the latinorigo, "beginning") in a coordinate system is the point where axes of the system intersect. The most common systems are two-dimensional (contained in a plane) and three-dimensional (contained in a space), having therefore two or three perpendicular axes. Axes are then divided in two halves from the origin, a positive and a negative one. This is usually done by defining a point of interest to us, and calling "positive side" the halves of the axes being closer to this point. Axes are usually called "X" and "Y" (in a plane) or "X", "Y" and "Z" (in a three-dimensional system).
All positions in the plane or space are then located in reference to the origin in terms of "5 distance units in direction of the positive half of the X axis" and "10 distance units in direction of the negative half of the Y axis" for a two-dimensional system, adding the part for the "Z" axis (for instance "12 distance units in direction of the negative side") in case of a three dimensional system. All this can be written in short as (5,-10,-12), which is the position of the point in reference to the origin, also known as the coordinates of the point. The coordinates of the origin are always (0,0) or (0,0,0), depending on the type of
Question: If a point is 15 units in the positive direction of the Y axis and 20 units in the positive direction of the Z axis, what are its coordinates in a three-dimensional system?
Answer: (0,15,20) | 677.169 | 1 |
Note that there are 7 points and 7 lines in this geometry. Since there are 7 points there are 21 pairs of points and you can check that there is always exactly one line which goes through this pair. For example, if one chooses the pair P6 and P7 the line which goes through them is L6. You can also check that each pair of lines meet in exactly one point. The lines above all lie along lines that exist in the Euclidean plane with one exception. This is the line labeled L6. This line looks like a circle. It can be proved that no matter how one tries to represent the Fano plane in the Euclidean plane that one can not represent all 7 of the liness in the Fano plane so that they lie along straight line. The best one can do is something like the diagram above, where one "line" does not lie along a Euclidean line.
Question: How many lines are there in the Fano plane? Answer: 7 | 677.169 | 1 |
Dihedral Figures
Use this activity to recognize dihedral symmetry and reflections in figures and examining various symmetries.
Instructions
All shapes in a figure are identical. The black shape is the seed shape. If you click and drag any of the red vertices of a black shape, it will alter both the black shape and all the identical blue shapes. For the best figures, try to keep the shapes from overlapping.
Exploration
A dihedral figure is one with rotational and reflectional symmetries. If you look at the top left image in the activity, you'll see it's labeled "Dihedral 2." This is because it has two symmetries, horizontal and vertical. The Dihedral 3 figure can be turned in 3 ways and reflected in 3 ways, and so forth.
Each figure in the activity has a different type of dihedral symmetry. Use the red vertices to change their shapes. As you alter the figures, try to answer these questions:
Are the figures always symmetrical?
Do the kinds of shapes used to make the figure determine the kind of symmetry?
Can you find all the rotational and reflectional symmetries?
Some figures have an odd dihedral symmetry, such as the one in the top right, while others have even dihedral symmetry. What do all odd dihedral figures have in common that's different from the even dihedral figures?
Question: How can you alter the shapes in the figures? Answer: By clicking and dragging any of the red vertices of a black shape. | 677.169 | 1 |
How to tell impossible length combination of triangle
I just answered this question but was unable to come up with some kind of formula to tell the user that some combinations of a triangle is impossible. Is there any kind of formula to tell if a combination of a triangle is impossible?
Re: How to tell impossible length combination of triangle
That link does not show the problem, so I deleted it to prevent confusion. Please copy the problem here. Then it can be answered.
I think I can deduce what you are asking.
For any combination of three lengths a,b,c this must be true to form a triangle.
a+b>c b+c>a a+c>b tell impossible length combination of triangle
Oh, sorry. The problem is:
Three towns are arranged in the shape of a triangle. The distance from Stockton to Overly is 10 miles and the distance from Overly to Shea is 8 miles. Maria drives from Stockton to Overly to Shea and back to Stockton. Which of the following statements best describes her trip?
i.imgur.com/6UUsh.png
The answers are: She drove 19 miles. She drove 36 miles. She drove less than 36 miles. She drove more than 36 miles.
Re: How to tell impossible length combination of triangle
Call Shea to Stockton d and then you know that d < 10 + 8. Therefore only answer c is possible.
You are a programmer
Question: Which of the following sets of lengths can form a triangle? (a) 10, 8, 18 (b) 10, 8, 10 Answer: (b) 10, 8, 10
Question: Which of the following is NOT a requirement for a set of three lengths to form a triangle? (a) The sum of the lengths of any two sides must be greater than the length of the third side (b) The sum of the lengths of any two sides must be equal to the length of the third side (c) The difference between the lengths of any two sides must be less than the length of the third side Answer: (b) The sum of the lengths of any two sides must be equal to the length of the third side | 677.169 | 1 |
Trigonometry
Introduction
In geometry we learn about how the sides of polygons relate to the angles in the polygons, but we have not learned how to calculate an angle if we only know the lengths of the sides. Trigonometry (pronounced: trig-oh-nom-eh-tree) deals with the relationship between the angles and the sides of a right-angled triangle. We will learn about trigonometric functions, which form the basis of trigonometry.
Investigation : History of Trigonometry
Work in pairs or groups and investigate the history of the foundation of trigonometry. Describe the various stages of development and how the following cultures used trigonometry to improve their lives.
The works of the following people or cultures can be investigated:
Cultures
Ancient Egyptians
Mesopotamians
Ancient Indians of the Indus Valley
People
Lagadha (circa 1350-1200 BC)
Hipparchus (circa 150 BC)
Ptolemy (circa 100)
Aryabhata (circa 499)
Omar Khayyam (1048-1131)
Bhaskara (circa 1150)
Nasir al-Din (13th century)
al-Kashi and Ulugh Beg (14th century)
Bartholemaeus Pitiscus (1595)
Note: Interesting Fact :
You should be familiar with the idea of measuring angles from geometry but have you ever stopped to think why there are 360 degrees in a circle? The reason is purely historical. There are 360 degrees in a circle because the ancient Babylonians had a number system with base 60. A base is the number at which you add another digit when you count. The number system that we use everyday is called the decimal system (the base is 10), but computers use the binary system (the base is 2). 360=6×60360=6×60 so for them it made sense to have 360 degrees in a circle.
Where Trigonometry is Used
There are many applications of trigonometry. Of particular value is the technique of triangulation, which is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. GPSs (global positioning systems) would not be possible without trigonometry. Other fields which make use of trigonometry include astronomy (and hence navigation, on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Discussion : Uses of Trigonometry
Question: What is the total number of degrees in a circle? Answer: 360 degrees | 677.169 | 1 |
You should find the angle A is 63,43∘63,43∘. For angle B, you first work out x (33,69∘33,69∘) and then B is 180∘-33,69∘=146,31∘180∘-33,69∘=146,31∘. But what if we wanted to do this without working out these angles and figuring out whether to add or subtract 180 or 90? Can we use the trig functions to do this? Consider point P in Figure 15. To find the angle you would have used one of the trig functions, e.g. tanθtanθ. You should also have noted that the side adjacent to the angle was just the x-co-ordinate and that the side opposite the angle was just the y-co-ordinate. But what about the hypotenuse? Well, you can find that using Pythagoras since you have two sides of a right angled triangle. If we were to draw a circle centered on the origin, then the length from the origin to point P is the radius of the circle, which we denote r. Now we can rewrite all our trig functions in terms of x, y and r. But how does this help us to find angle B? Well, we know that from point Q to the origin is r, and we have the co-ordinates of Q. So we simply use the newly defined trig functions to find angle B! (Try it for yourself and confirm that you get the same answer as before.) One final point to note is that when we go anti-clockwise around the Cartesian plane the angles are positive and when we go clockwise around the Cartesian plane, the angles are negative.
So we get the following definitions for the trig functions:
sinθ=xrcosθ=yrtanθ=yxsinθ=xrcosθ=yrtanθ=yx
(20)
But what if the x- or y-co-ordinate is negative? Do we ignore that, or is there some way to take that into account? The answer is that we do not ignore it. The sign in front of the x- or y-co-ordinate tells us whether or not sin, cos and tan are positive or negative. We divide the Cartesian plane into quadrants and then we can use Figure 16 to tell us whether the trig function is positive or negative. This diagram is known as the CAST diagram.
Figure 16
We can also extend the definitions of the reciprocals in the same way:
cosecθ=rxsecθ=rycotθ=xycosecθ=rxsecθ=rycotθ=xy
(21)
Exercise 4: Finding the angle
Points R(-1;-3) and point S(3;-3) are plotted in the diagram below. Find the angles αα and ββ.
Figure 17
Solution
Step 1. Write down what is given and what is required :
Question: What is the relationship between the x-coordinate and the side adjacent to the angle in a trigonometric function?
Answer: The x-coordinate is the length of the side adjacent to the angle.
Question: What is the measure of angle A?
Answer: 63.43 degrees | 677.169 | 1 |
angle symbol
The angle symbol is a mathematical symbol that is placed ahead of character s, usually uppercase italic letters representing spatial points, to describe a geometric angle formed by the intersection of two lines, line segments, or rays. The symbol looks like a skewed, uppercase, sans serif letter L ( ).
Next Steps
The illustration shows an example of the use of the angle symbol (along with some other mathematical symbols) in a geometric scenario. The statement at the bottom translates to, "Lines L and M are parallel if and only if angle ABC is congruent to angle DEF ."
When two angles are congruent, they have equal measure, so an equals sign (=) is sometimes used in place of the congruence symbol. Thus we can say that in this example, ABC = DEF
Question: What symbol is used to denote that two angles are congruent? Answer: The symbol for congruent angles is the Greek letter "∠" or the abbreviation "cong". | 677.169 | 1 |
Question 8068
I'll show you how to do one of them and you can do the others.
You can derive the formula yourself by doing the following:
1) In any polygon, from one vertex (corner), draw all the diagonals to the other vertices.
2) Now count the number of triangles you have made inside the polygon. This number will always be 2 less than the number of sides in the polygon.
Each triangle has a total of 180 degrees as the sum of its interior angles.
3) Multiply the number of triangles you made by 180 degrees and this will be the sum of the interior angles of the polygon.
The formula is:
(n-2)180 degrees. Where n is the number of sides of the polygon.
Let's do the 20-gon, (icosagon?)
The number of sides is 20, so n = 20.
(n-2)180 degrees = (20 - 2)180 = (18)180 = 3240 degrees.
You can do the others using the same formula.
Question: How many triangles are formed when drawing diagonals from one vertex of a polygon to the other vertices? Answer: 2 less than the number of sides in the polygon. | 677.169 | 1 |
conchoid
con·choid
a plane curve such that if a straight line is drawn from a certain fixed point, called the pole of the curve, to the curve, the part of the line intersected between the curve and its asymptote is always equal to a fixed distance. Equation: r = b ± a sec(θ).
geometry a plane curve consisting of two branches situated about a line to which they are asymptotic, so that a line from a fixed point (the pole) intersecting both branches is of constant length between asymptote and either branch. Equation: (x -- a)²(x² + y²) = b²x² where a is the distance between the pole and a vertical asymptote and b is the length of the constant segment
Question: What is the special property of the curve regarding a line drawn from a fixed point (the pole) to the curve? Answer: The part of the line intersected between the curve and its asymptote is always equal to a fixed distance. | 677.169 | 1 |
Like the hyperbola and parabola, the ellipse is a conic section.
Related article: Conics
empty set
The unique set having no elements, generally denoted by a circle with a forward slash through it, or by an empty pair of braces.
Epimenides Paradox
See Liar Paradoxilateral triangle
A triangle with equal sides and equal angles.
uler number
The transcendental numbere, approximately 2.71828.... It may be defined as the limit
The functionex is called the exponential function, and has the property that it is its own derivative.
Question: What is the property of the exponential function e^x? Answer: It is its own derivative. | 677.169 | 1 |
This geometry
study is a continuation of the analysis of the Manton Drove Crop Circle.
While this study does not offer new insight for the polar clock's
date/time, the design maintains focus on the role of Pi in this crop
circle geometry.
This geometry promotes the theory that a squared circle can be proven if
these dimensions are reflected in two adjoining sides of a geometric
object: 1/2 the square root of Pi and 1/2 the square root of 2. The
inscribed tan quadrilateral, named "Karotumus" (for its tasty discovery;
accent on the second syllable), appears to include these dimensions.
Intuitively, the 1000-units-diameter blue corral (the circle) should
limit the infinity of Pi, but does it? .... or is this the "Enigma of
Karotumus"?
(Focus on the circle squaring properties of the right triangle
within Karotumus: angles = 62.402.., 27.597.., 90 degrees.)
This new concept of Pi will intrigue geometers as well as crop
circle aficionados. Please add the following comment after the
sentence "Focus on the circle squaring properties ...":
Consider also that the Pythagorean Theorem confirms these
angles:
(1/2 square root of Pi = 0.88622692545275801364908374167057...)
a = 463.25137517610424292137983379471... (top horizontal
side)
b = 886.22692545275801364908374167057... (left vertical
side)
c = 1000 units (hypotenuse; radius of large magenta circle)
a˛ + b˛ = c˛ = 999999.99999999999999999999999999...
Here's the latest research on the Manton
Drove crop circle's geometry as it relates to the Barbury
Castle's Pi crop circle (June, 2008).
I was curious about the rounding of Pi
(from 3.1415926535... to 3.141592654...) in the Pi crop circle
since I believed that a number is not rounded if an ellipsis is
used to continue that number. But Bert Janssen's conjecture on
those "three little circles" in the Pi crop circle provided
great insight: that ellipsis may also convey circle squaring
information! (see:
)
Interestingly, the circle makers clearly
demonstrate their geometry expertise by choosing this tenth
Question: What is the value of 'a' in the Pythagorean theorem for this triangle? Answer: a = 463.25137517610424292137983379471...
Question: What are the angles of the right triangle within Karotumus? Answer: The angles are approximately 62.402°, 27.597°, and 90 degrees.
Question: What is the name of the inscribed quadrilateral in this geometry? Answer: Karotumus.
Question: What are the two dimensions that the study suggests could prove a squared circle? Answer: 1/2 the square root of Pi and 1/2 the square root of 2. | 677.169 | 1 |
Sunday, September 16, 2012
171: Something Euclid Missed
Before we start, a quick reminder: if you like the podcast, don't forget to go to iTunes and post a good review! I know, I promised not to be one of THOSE podcasters, but I just noticed that Math Mutation has fallen off the iTunes "What's Hot" list of top Science and Medicine podcasts, and it would be nice to get back up there. Anyway, on to today's topic.
Here's an experiment for you. Take a piece of paper, and try using your abundant artistic skills to drawa triangle. Any triangle will do: it can be equilateral, isoceles, right, or none of the above, just the lines connecting three random dots on your paper. Now trisect each angle: draw lines 1/3 and 2/3 of the way across the angle at each corner. There should be three points near the middle of the triangle where pairs of adjacent trisectors intersect each other. Join these to form a little triangle in the center. Assuming you are sufficiently talented to be able to draw straight lines, you will notice something amazing: no matter what triangle you started with, the small one in the middle is equilateral, with all sides the same length and all angles at 60 degrees! How did this happen?
This little equilateral triangle is known as Morley's Triangle. You would think that such a simple trick, drawing lines within a triangle to get cool shape in the middle, would have been part of the classical geometry known since Euclid, but surprisingly, that's not the case. Euclid and his contemporaries may have missed this due to his tendency to concentrate on figures that could be constructed with compass and straightedge, since trisecting angles isn't directly possible with this type of technique. This triangle wasn't discovered until 1899, by Anglo-American mathematician Frank Morley at Haverford College. Morley was actually investigating complex properties of more general algebraic curves, and came across this triangle by accident. He didn't bother publishing it right away, though it spread by word-of-mouth until it eventually appeared in print as a problem in The Educational Times in 1908. He also showed it to his young son, who was fascinated by this magic triangle and later reminisced, "Always, to the eye at least, the theorem, if drawn accurately, proved itself. What caused me considerable annoyance was that I could not for a long time comprehend what purblind examiners might accept as a valid proof. " These recollections also hint that part of Morley's reluctance to publish may have come from the fact that the theorem seemed so simple and obvious (once drawn) that he was sure somebody must have already discoved it centuries ago. But the first actual publication of a proof was by two other mathematicians named Taylor and Marr in 1913, who acknowledged Morley in their paper.
Question: Who discovered Morley's Triangle? Answer: Frank Morley
Question: What was Frank Morley's profession? Answer: Mathematician | 677.169 | 1 |
Since Morley, numerous proofs have been discoverered of the theorem. Trying to guess at the intuition behind the Morley triangle, it occurred to me that 180 degrees is a special quantity for triangles, the sum of their angles, so it only makes sense that when messing around with trisected, or 1/3, angles, the quantity 60 degrees, which matches the angles at the corner of an equilateral triangle, would play a special role. Unfortunately, I haven't been able to find a proof that really connects to this intuition as to why this magical equilateral triangle appears. Many proofs are basically solving a set of trigonometric equations to figure out the relations of the lines and angles, fully valid and convincing but not providing much insight. Probably the cleanest proof I've seen online is one discovered in the late 20th century by Conway, where he basically assembles a bunch of small triangles with the right sides and angles, and shows how they fit together to form any larger triangle with an equilateral Morley triangle in the center. You can see nice illustrations of this at the links in the show notes.
But an even more surprising aspect of Morley's theorem is that it can be generalized to find other implied equilateral triangles lurking around. We've been talking about trisecting the interior angles of a triangle-- but what about the *exterior* ones? Actually, if you draw the exterior trisectors of each angle of the triangle, you can come up with yet more equilateral triangles, both by intersecting the exterior trisectors with each other, and intersecting interior and exterior trisectors. You can also come up with slightly different trisectors by adding 360 or 720 degrees to the size of an angle and then dividing by three, yielding yet more implied triangles. There are a total of 18 Morley triangles that can be constucted. One amusing article on the net, linked in the show notes, is from a math enthusiast who wrote a computer program trying to illustrate the central Morely triangle we started with, but due to a bug actually trisected the exterior angles in some cases... and was surprised to produce a equilateral Morley triangles anyway!
I think the coolest aspect of this whole Morley triangle concept is that we had a supposedly well-explored, solidly understood area of mathematics, the Euclidean geometry of planar triangles, and thousands of years later a new and unknown property was discovered. Just draw the trisectors of each corner of any triangle, and their points of intersection determine an equilateral triangle. The ancient Greeks could have discovered the Morley triangle and come up with a proof like Conway's, but somehow they didn't, despite some of them having a literally religious devotion to geometry. How many more surprises are lurking in what we today consider well-understood areas of math? Maybe someday a Math Mutation listener will be the one who discovers something new. Maybe even you.
2 comments:
Good suggestion-- except that video podcasts take a lot more work. If I win the lottery & am able to quit my day job, maybe I'll start doing that. :-) Meanwhile, you can find some good diagrams at the links above.
Question: What is the sum of the angles in a triangle? Answer: 180 degrees
Question: What happens when you trisect the exterior angles of a triangle? Answer: You can come up with more equilateral triangles by intersecting the exterior trisectors with each other and with interior trisectors.
Question: What is the special role of the quantity 60 degrees in the context of Morley's theorem? Answer: It matches the angles at the corner of an equilateral triangle and plays a special role in the theorem. | 677.169 | 1 |
Tuesday, September 4, 2012 at 9:28pm
Geometry Find m∠T if m∠T is 20 more than four times its supplement.
Tuesday, September 4, 2012 at 6:50pm
ap geometry m is the midpoint of segment jk. jm=x/8 and jk=3x/4 subtracted by 6. find mk
Monday, September 3, 2012 at 11:10pm
geometry a Street in san franscico has a 20% grade-that is it rises 20' for every 100' horizontally. Find the angle of elevation of the street?
Monday, September 3, 2012 at 7:073231pm
Geometry Draw a figure demonstrating a quadrilateral that is not a parellelogram such that diagonals are congruent and perpendicular.
Sunday, September 2, 2012 at 8:06am
geometry construct/locate the orthocenter of a triangle LMN where LM=7 cm <M=130 and MN=6 cm and its process
Sunday, September 2, 2012 at 5:40am
Geometry Honors (p.2) Two planes that lie in parallel planes are ________. Answers: Always, Never, or Sometimes I'm having some trouble on this one... I think it's sometimes...?
Thursday, August 30, 2012 at 8:29pm
Geometry Honors is XY (with a --> over it) equal to YX (with a --> over it) in other words, the ray XY and the ray YX... Can anyone please clarify this for me...?
Thursday, August 30, 2012 at 8:25pm
Geometry Find, the nearest tenth, the area of the region that is inside the square and outside the circle. The diameter of the circle is 14in.
Thursday, August 30, 2012 at 11:17am
Math In euclidean geometry, the sum of the measures of the interior angles of a pentagon is 540. predict how the sum of the interior angles of a pentagon would be different in spherical geometry
Thursday, August 30, 2012 at 7:53am
geometry I need help on this question "Find the value of the variable and GH if H is between G and I". GI=8b+2,HI=3b-2,HI=16 Please helo me....
Wednesday, August 29, 2012 at 8:53pm
geometry If F is between E and G, and EF = 8x - 14, FG = 4x -1 and EG =59, then what does x equal?
Wednesday, August 29, 2012 at 7:22pm
Question: What is the length of MK if M is the midpoint of segment JK, JM = x/8, and JK = 3x/4 - 6? Answer: MK = (3x/4 - 6)/2 = (3x - 12)/8
Question: How would the sum of the interior angles of a pentagon differ in spherical geometry compared to Euclidean geometry? Answer: The sum of the interior angles in spherical geometry would be less than 540 degrees due to the curvature of the sphere. | 677.169 | 1 |
Tuesday, August 21, 2012 at 7:42pm
Geometry You want to frame a picture that is 5 inches by 7 inches with a 1 inch wide frame. What is perimeter of outside edge of the frame.?
Tuesday, August 21, 2012 at 11:05am
geometry In Exam Figure 7, AA′ = 33 m and BC =7.5 m. The span is divided into six equal parts at E, G, C, I, and K. Find the length of A′B. Round your answer to two decimal places.
Sunday, August 19, 2012 at 11:53am
Geometry If a circle with center O has equilateral triangle AOB inscribed with sides 10 inches long. If OC is perpendicular to AB at P, what is the length of AP in inches?
Saturday, August 18, 2012 at 4:08pm
geometry The sides of a quadrilateral are 3, 4, 5, and 6. Find the length of the shortest side of a similar quadrilateral whose area is 9 times as great.
Friday, August 17, 2012 at 9:42am
geometry Campsite F And G Are On Opposite Sides Of The Lake.A Survey Crew Made Measurements Shown On The Diagram .What Is The Distance Between The Two Campsites?
Friday, August 17, 2012 at 5:37am
geometry Prove that median of a isoeceles trinangle is perpendicular to the unequal sides.
Thursday, August 16, 2012 at 9:15am
geometry ABC IS a isoceles triangle in which AB=AC 'IF AB and AC are produced to D and E respectively such that AB = CE.prove that BE =CD
Wednesday, August 15, 2012 at 12:32pm
geometry Is it possible for two points on the surface of a prism to be neither collinear nor coplanar?
Tuesday, August 14, 2012 at 8:37pm
geometry what are the greatest and least possible lengths to make necklaces from a 14 3/4 inch length of a cord
Tuesday, August 14, 2012 at 7:35pm
Geometry Finish pattern George John Thomas John
Monday, August 13, 2012 at 11:02pm
Geometry Factor to solve the equation 3x^2+10x-98=0
Monday, August 13, 2012 at 10:37pm
Geometry Suppose J is between H and K. Use the Segment Addition Postulate to solve for x. Then find the length of each segment. HJ=2x+5 JK=3x-7 KH=18
Question: What are the greatest and least possible lengths to make necklaces from a 14 3/4 inch length of a cord? Answer: Greatest length - 14 3/4 inches, Least length - 7 1/4 inches
Question: What is the length of AP in the equilateral triangle AOB with sides of 10 inches, where OC is perpendicular to AB at P? Answer: 5 inches
Question: What is the length of the shortest side of a similar quadrilateral whose area is 9 times as great, given the sides of the original quadrilateral are 3, 4, 5, and 6? Answer: 3 inches
Question: What is the factor to solve the equation 3x^2 + 10x - 98 = 0? Answer: -7 or 14 | 677.169 | 1 |
sin45=1/√2 cos45=2/√2 tan45=1
sin could also be expressed as √2/2=sin45
Then there is the 30-60-90 triangle:
in this case, it would be the following:
sin30=0.5 sin60=√3/2
cos30=√3/2 cos60=1/2
tan30=√3/3 tan60=√3/1
Non-Right Triangles:
In cases where you don't know if the triangle has a right angle you must use the law of sines:
SinA/a=SinB/b=SinC/c
Also to help you to make a right triangle when there doesn't appear to be one is to cut the triangle down the center as shown in the picture to the left. Then you have two right triangles to deal with. :-)
But in the example to the right it is undefined due to the 37 being an impossible number on the side that it is on due to the rest of the triangle.
Now lets look at some examples using the law of sines. :-D
So with this example on the right you would do the following:
Sin25/15=SinX/26
26sin(25)/15=x
Sin^-1(26sin25/15)=47.099
X=47.099
Law of cosines:
The law of cosines is:
c^2=a^2+b^2-2ab·cosC
Now that you know this lets look at some examples:
5^2=7^2+9^2-2(7)(9)cosY
25=49+81-2(63)cosY
25=130-126cosY
25-130=(130-130)-126cosY
-105=-126cosY
-105=(-126/-126)cosY
-105/-126=cosY
0.8333=cosY
cos^-1(0.8333)=Y
Y=33.56
AND REMEMBER DON'T COMBINE TERMS THAT YOU THINK ARE ALIKE, BUT REALLY
AREN'T!!
Next is another example:
x^2=12^2+8^2-2(12)(8)cos25
x^2=208-2(96)cos25
x^2=208-192cos25
x^2=33.9889
x=√33.9889
x=5.83
But there is also another way to write the law of cosines:
Question: In the example with a 25-degree angle and sides of 15 and 26, what is the measure of angle X? Answer: 47.099 degrees
Question: What does the law of sines state? Answer: SinA/a = SinB/b = SinC/c
Question: What is the value of sin45? Answer: 1/√2 | 677.169 | 1 |
The logical structure of the exposition of the proofs has influenced all scientific thinking since Euclid's time.
This logical structure is essentially as follows:
A statement of the proposition.
A statement of the given data (usually with a diagram).
An indication of the use that is to be made of the data.
A construction of any additional lines or figures.
A synthetic proof.
A conclusion stating what has been done.
In about 300 � Euclid established a set of axioms for geometry.
One of them assumes the existence of one and only one line
that is both parallel to a given line and contains a given point
that is not a point of the line; or,
given a line and a point not on the line, one and only one coplanar line can be drawn through the point parallel to the given line."
Non-Euclidean geometres have been developed by denying the validity of
the famous fifth postulate (parallel postulate),
and are based on alternatives to it.
The discovery of non-Euclidean geometries inspired a new approach to
the subject by presenting theorems in terms of axioms applied to
properties assigned to undefined elements called points and lines,
not necessarily limited to the classical study of flat surfaces
(plane geometry) and rigid three-dimensional objects (solid geometry).
This led to many new different types of geometries such as, for example,
hyperbolic geometry, projective geometry, Riemannian geometry, elliptical, and parabolic geometries.
Trigonometry
Trigonometry is the study of triangles, angles, and specific functions
of angles and their application to calculations in geometry.
Trigonometry is the branch of mathematics used in computing,
rather than directly measuring, distances.
The trigonometric functions
(sine, cosine, tangent, cotangent, secant, and cosecant)
can be defined as the ratios of lengths of sides of right triangles,
or in terms of coordinates of terminal points of arcs on the unit circle
(circular functions).
For example,
if a right-angled triangle contains an angle, symbolized here
by the Greek letter alpha, α,
the ratio of the side of the triangle opposite
to α to the side opposite the right angle (the hypotenuse)
is called the sine of α.
The ratio of the side adjacent to α
to the hypotenuse is the cosine of α.
These functions are properties of the angle α,
and calculated values have been tabulated for many angles.
These are useful in determining unknown angles and distances
from known or measured angles in geometric figures.
The subject developed from a need to compute angles and distances in
such fields as astronomy, map making, surveying, and artillery range
finding.
Question: What is the sine of an angle in a right-angled triangle? Answer: The ratio of the side of the triangle opposite the angle to the side opposite the right angle (the hypotenuse)
Question: What does the parallel postulate assume? Answer: The existence of one and only one line that is both parallel to a given line and contains a given point not on the line | 677.169 | 1 |
Math Ed Blog from Bruce Yoshiwara
Thursday, March 28, 2013
The angle bisectors
of the triangle meet at the center of the inscribed circle of radius r. If we let \(2\alpha=A\), \(2\beta = B\), and \(2\gamma=C\), we have \(\alpha+\beta+\gamma=\frac{\pi}{2}\).
Let x be the distance from the vertex at A to points of tangency, y the distance from B, and z the distance from C. Then then lengths of the triangle sides opposite A, B, and C are respectively \(a=y+z\), \(b=x+z\), and \(c=x+y\).
Thus if we name the semiperimeter s, then \(s=x+y+z\), \(x=s-a\), \(y=s-b\), and \(z=s-c\).
The radii at the
points of tangency and the angle bisectors form 3 pairs of congruent
triangles. The area of \(\Delta ABC\) is \(xr+yr+zr= r(x+y+z)\), so area \(=rs\), and \( (\text{area})^2=r^2s^2\). Using results we have above, we
obtain
\[ (\text{area})^2 = s\cdot xyz = s(s-a)(s-b)(s-c)\]
so the area is \(\sqrt{s(s-a)(s-b)(s-c)}\).
Thursday, December 20, 2012
A primary goal of
the Common Core State Standards (CCSS) is to provide a curriculum to ensure
that all high school graduates are college and career ready. The CCSS math topics through grade 11 include not only
all of the topics of the traditional U.S. Algebra 1-Geometry-Algebra 2
sequence, but also topics typically taught in courses named trigonometry and
statistics.
Alternative pathways
provide a means for non-STEM (i.e., non- Science, Technology, Engineering, and
Math) students to transfer from a two-year college to a four-year institution
and earn a bachelor's degree without needing to show mastery of traditional intermediate
algebra topics. The promotion of alternative pathways challenges the premise
that the CCSS for math are needed for all
students to be college ready.
The common goal of both alternative pathways and the CCSS is to improve U.S. education.
"Core Principles for Transforming Remedial Education: A Joint Statement" from the Charles A. Dana Center,
Complete College America, Inc., Education Commission of the States, and Jobs
for the Future, calls for revamping the two-year college remediation
structure. The paper lists seven Core
Principals for a "fundamentally new approach for ensuring that all
students are ready for and can successfully complete college-level work that
leads to a postsecondary credential of value.
"...Principle 2.
The content in required gateway courses should align with a student's academic
program of study — particularly in math.
"Gateway
courses provide a foundation for a program of study, and students should expect
that the skills they develop
Question: What is the primary goal of the Common Core State Standards (CCSS) in mathematics? Answer: The primary goal of the CCSS in mathematics is to ensure that all high school graduates are college and career ready.
Question: Which organizations collaborated to create the "Core Principles for Transforming Remedial Education" paper? Answer: The Charles A. Dana Center, Complete College America, Inc., Education Commission of the States, and Jobs for the Future collaborated to create the "Core Principles for Transforming Remedial Education" paper.
Question: What is the relationship between the sum of the angles of a triangle and the sum of the angles bisectors? Answer: The sum of the angles bisectors (α, β, γ) of a triangle is equal to π/2, i.e., α + β + γ = π/2. | 677.169 | 1 |
Common Error #17 Student Work Samples
This student did not correctly graph the ordered pairs.
This student correctly plotted the given ordered pairs and found the midpoints of the segments.
Page 33 9/14/2012
Common Error #18
Students do not correctly draw polygons, both regular and irregular.
A homework problem in Tao's geometry book asks him to draw an isosceles triangle in which at
least one of the angles is 40. One of Tao's friends claims that there are two answers to this
question, because there are two different triangles which can be drawn that meet the criteria. Draw
both triangles and label the angle measures.
Page 34 9/14/2012
Common Error #18 Student Work Samples
The student correctly drew the first triangle, but did not make the second triangle isosceles.
Both triangles are correctly drawn, marked and labeled.
Page 35 9/14/2012
Common Error #19
Students do not draw a reflected figure the same distance from the line of reflection as the original
figure, as well as preserve the shape and size of the original figure.
Draw a horizontal or vertical line of your choosing on the coordinate plane below, and draw the
reflection of the triangle over your line.
Draw a non-vertical or non-horizontal line of your choosing in the space below, and draw the
reflection of the figure over your line.
Page 36 9/14/2012
Common Error #19 Student Work Samples
Incorrect reflection.
Correct reflection.
Page 37 9/14/2012
Common Error #20
Students need practice sorting figures using more than one attribute (example: four-sided figures
with exactly one line of symmetry). When sorting figures with specific attributes, students
mistakenly assume that there must be an equal number of figures for each attribute. Give students
practice situations where the sorting does not result in equal numbers of figures in each category.
Sort each of the geometric figures below into three groups based on the attributes of each figure.
Describe the criteria you are assigning each group. When deciding on your criteria, you may not
have a figure which fits into two different groups.
Group I Group II Group III
Criteria: Criteria: Criteria:
Figures: Figures: Figures:
Page 38 9/14/2012
Common Error #20 Student Work Samples
This student did not create three mutually exclusive groups. All of the items in Group II may also
be put in Group III.
Page 39 9/14/2012
Common Error #20 Student Work Samples, continued
This student correctly sorted the figures into groups.
Page 40 9/14/2012
Common Error #21
Students need to understand measures of central tendency: mean, median, and mode.
Wes and his six children are playing in a carnival "house of mirrors". The weights of the seven
Question: Which student correctly sorted the figures into three mutually exclusive groups? Answer: The student on Page 39.
Question: What is the task given to the students in Common Error #20? Answer: To sort each of the geometric figures into three groups based on their attributes. | 677.169 | 1 |
rotating lines in 2d space
Hi,
I'm struggeling with with a problem and culd use some help.
I have 2 lines somewhere in 2D space. One line is described by two known points p1, p2.
The other line is described by the unknown points p3, p4, but with a known length.
Now I need to connect p3 of the second line to either p1 or p2. The direction/alignment of the second line is defined by a known angle between the first line.
My problem now is to determine the second point p4 of the second line, given by p1, p2, p3 connected with either p1 or p2 and the angle between those lines.
I guess I have to apply some sort of rotation but I can't figure out how.
Question: What is the main task or problem the user is trying to solve? Answer: The user wants to determine the second point p4 of the second line, given the known points and the angle between the two lines. | 677.169 | 1 |
Re-arranging SOHCAHTOA
'Understand all the things that are true about a given diagram, choose the right one and re-arrange it to suit you!'
Once you know about SOHCAHTOA the next challenge is to find out how to apply it to different problems. After some practice it gets easier to figure out if you need Sin, Cos or Tan, but in the beginning it can be difficult. In any case, you can always make a statement/equation involving each of them. When you have done that you can see which is the most useful for solving the problem you have. Once you have done that, you need to re-arrange the equation to suit your purposes. Frustratingly, with all the trigonometry done, this can be the more difficult bit! It shouldn't be though and this activity hopes to help remember the nature of this relationship between three variables.
Resources
The first part of the activity involves the Making Statements worksheet. The second involves the Re-arranging SOHCAHTOA presentation which students are to re-arrange! Students can do this on their own computers or it can be done from a central computer as a whole class activity.
Making Statements
Description
Students can do this on their own computers or it can be done from a central computer as a whole class activity.
Students are given the 'Making Statements' task in which they are invited to write down equations that can model any of the variables in the diagram.
Students need access to computers to open the 'Re-arranging SOHCAHTOA', where they are invited to move the objects on each slide around to express relations in different ways.
I did it my way!
As a practising maths teacher I know that most us like to give activities our own little twist and do them 'our way'. It would be great to add a little collection of 'twists' from users. You can either add your twist to the comments section below or e-mail them directly to me at [email protected] In time some of these twists may appear here....
Question: What is the main challenge after understanding SOHCAHTOA? Answer: The main challenge is to apply the knowledge to different problems and figure out which trigonometric ratio (sin, cos, or tan) to use.
Question: What is the first part of the activity called? Answer: The first part of the activity is called 'Making Statements'. | 677.169 | 1 |
Star of David
A. Bogomolny
Let's start with a nice configuration I have referred to on a couple of occasions. A billiard ball has been kicked at a 60o angle to a side of a billiard table in the form of an equilateral triangle. After bouncing twice from each side, the ball eventually returns to its starting point. The distance travelled does not depend on the starting point, except for one case. If the ball starts in the middle of a side, then it returns to that point only after a three legged journey, which is only half as long as for other starting points. If we consider the length of the trajectory between two traversals of the starting point as a function of that point, we get an example of a naturally discontinuous function.
From a geometric viewpoint, we are given a hexagon inscribed into an equilateral triangle. What can be said about that hexagon? By construction, at vertices, its sides are equally inclined to the sides of the triangle.
This reminds us of the mirror property enjoyed by the orthic triangle. May we then generalize the original configuration to an arbitrary triangle? Yes, this is possible if we draw lines parallel to the sides of the orthic triangle.
This procedure generates a closed curve — a hexagon or a triangle — whose length is a discontinuous function of its position as defined by, say, any of its vertices. (Observation: the sides of the hexagon intersect on the altitudes of the triangle. When three sides are concurrent, they intersect at the orthocenter of the triangle.)
We are not done yet. The original configuration permits one other description. For example, we may notice that the sides of the hexagon are parallel to the sides of the given equilateral triangle. This construction can also be generalized:
Drawing lines parallel to the sides of an arbitrary triangle we also get a closed curve, a hexagon to be exact, except for the case when we start at a mid point of a side, in which case we obtain the medial triangle. And the length of the curve is again a discontinuous function of the starting point. (Observation: the sides of the hexagon intersect on the medians of the triangle. When three sides are concurrent, they intersect at the centroid of the triangle.)
Well, starting with an absurd billiard table we obtained two apparently unrelated configurations. In one, the inscribed hexagon has its sides parallel to the sides of the orthic triangle. In the other case, the sides of the hexagon are parallel to the sides of the given triangle. However, there is a little more to it. The orthic triangle has an interesting property. The medial triangle cuts from the given one three triangles similar to the given triangle. This is obvious as the triangles have parallel sides. Interestingly, the orthic triangle has exactly same property. It cuts off three triangles similar to the given one.
Question: What shape is formed by the points where the sides of the hexagon intersect when they are parallel to the sides of the given triangle? Answer: A triangle (specifically, the medial triangle)
Question: How many times does the ball bounce off each side of the table before returning to its starting point? Answer: Twice | 677.169 | 1 |
Take a point in the plane of a triangle and the Cevians through that point. There exist two triangles inscribed into the given one with sides parallel to the Cevians. When the Cevians are altitudes, the triangles combine into a Tucker hexagon inscribed into the second Lemoine circle. In the general case, the hexagon is no longer cyclic, but the triangles are still equal and centrally symmetric.
Hidefumi Katsuura from San Jose State University invented an iterative process that leads to that configuration. Start with a point (click anywhere in the applet) and draw a perpendicular to a side of a given triangle. From there drop a perpendicular to another side and so on. Putting it a little differently, the process requires us to draw lines parallel to the altitudes of a given triangle. This can be accomplished in two ways, depending on the ordering of the altitudes. Both processes converge, each to a triangle. And not surprisingly, these are exactly the triangles we just met above. The story was described by D. Gale in The Mathematical Intelligencer and subsequently in his book.
H. Katsuura has noted that the resulting configuration of the two limit triangles resembles the Star of David, albeit a distorted star when the given triangle is not equilateral.
Of course there is an interesting generalization. Two iterative processes may be run for any three concurrent Cevians. However depending on the point of concurrence, the iterations may or may not converge.
Jean-Pierre Ehrmann — a French member of the Hyacinthos discussion group — produced the following diagram where the region of divergence is marked in blue.
Related to the above are the following two exercises.
This one is relatively easy. From the base vertices, draw two arbitrary lines to the sides opposite the vertices. From there draw two additional lines parallel to the previous ones. The two points of intersection of the latter with the sides of the triangle lie on a line parallel to the base.
The second one is difficult. Let's make it a math droodle. What is it? The combination of the two proves that, in the presence of convergence, the limit triangles are always equal.
Question: What does the iterative process involve? Answer: The process requires drawing lines parallel to the altitudes of a given triangle in a specific order. | 677.169 | 1 |
The word "secant" comes from the Latin word for "cut", which came from the Indo-European root "sek", meaning "cut". The IE root also came directly into English via various Germanic sound changes to give us "saw" and "sedge".
The picture
Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept. The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points. You also get a very strong understanding of how the secant line is a function of the two given points. I don't think that is obvious to someone without some experience with such things.
This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects. (Math books are full of such pictures.) So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds. This is the sort of claim that is amenable to field testing.
The metaphor
Most metaphors are based on a physical phenomenon. The mathematical meanings of "secant" use the metaphor of cutting. When the word "secant" was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor. In those days essentially every European scholar read Latin. To them "secant" would transparently mean "cutting". This is not transparent to many of us these days, so the metaphor may be hidden.
If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.
The straight line does not really cut the curve. Indeed, the curve itself is both an abstract object that is not physical, so can't be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it? Cut the screen? The line can't do that.
You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
The metaphor is restricted further by saying that it is determined by two points on the curve. This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines. You could define such a family by using one point on the curve and a slope, for example. This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.
Secant on circle
Another use of the word "secant" is the red line in this picture:
Question: Which language was "secant" transparently understood in its metaphorical meaning? Answer: Latin.
Question: What is the origin of the word "secant"? Answer: The word "secant" comes from the Latin word for "cut", which in turn comes from the Indo-European root "sek", meaning "cut".
Question: What is another use of the word "secant" in mathematics? Answer: The red line in the picture of a circle.
Question: True or False: The straight line in the "secant" metaphor actually cuts the curve. Answer: False. The straight line does not physically cut the curve. | 677.169 | 1 |
The ideas in this geometry lesson are taken from the Geometry ebook that I sell at MathMammoth.com.
This lesson plan does not contain all the problems the Geometry ebook
does.
Measuring angles Free geometry lesson plan from HomeschoolMath.net
Angles are measured in degrees.
Remember how you can picture the one side of the angle as tracing out a
circle or an arc of a circle. The FULL CIRCLE forms a 360 degree
angle. Therefore a half circle or a straight angle is 180 degrees, and a
fourth of a circle or a right angle is 90 degrees. Look at the
pictures. We use the little circle
to denote the degree after numbers.
This is a 1 degree angle!
This is a protractor. It is used to measure angles.
Note how it has the shape of half a circle;
therefore it only measures angles up to 180.
It has two sets of numbers: one set goes
from 0 to 180 one way, one set from 0 to 180
the other way. Which one you read depends on
where you place the one side of the angle you
are measuring.
[Picture available in the ebook]
[Picture available in the ebook]
To measure an angle, place the little circle or open hole of the
protractor on the VERTEX of the angle. Place the zero line of the
protractor on the ONE SIDE of the angle. Then read the measure
where the other side hits the protractor scale. This angle is obviously
an OBTUSE angle, so we read the scale at 127 degrees.
These pictures illustrate
how to measure angles from triangles or other figures.
The zero line of the protractor needs to be lined up with one side of
the angle, and you read the set of numbers from your zero line on.
[Other pictures available in the ebook]
To draw an angle of
50,
first draw a line
segment that is to be
the one side of the angle.
Then put the protractor so that its
zero line matches with your line
segment and that the vertex is in
place. Then put a little mark at the
50
spot.
Take the protractor off
and draw a line through
your mark.
[Pictures available in the ebook]
These pictures illustrate drawing a 70
angle so that it shares a side with an existing angle.
Example problem types
1. Measure the following angles. If necessary, continue the sides
of the angle..
.
2. Measure all the angles in these triangles.
3. Draw an angle of a) 35
b) 76 c) 137 d) 162
4. Here you see a pie with angles of 90,
75, 75, and 120 . Draw into the other pie diagram angles of 35
and 160 degrees. The picture already has a right angle. What
Question: What is the unit used to measure angles? Answer: Degrees | 677.169 | 1 |
Symmetry can be found all around us. You need only look in the mirror to find evidence that the human body is symmetrical. If you draw a line down the middle of your body, you will see that one side is the mirror image of the other side: two eyes, two ears, two arms, two legs, ten fingers, etc.
The type of symmetry first mentioned in the video segment is called bilateral symmetry, which is also called reflection symmetry or mirror symmetry. A figure has bilateral symmetry if you can draw a vertical or horizontal line through the middle of the figure and one half of the figure is a mirror image of the other half. The line down the middle is called a line of symmetry. In the video segment, the first leaf shown by the plant expert exhibits bilateral symmetry.
Another common type of symmetry is rotational symmetry. If a figure can be rotated a certain number of degrees about its center and look exactly the same, the figure is said to have rotational symmetry. The angle through which the figure can be rotated to make it look the same is called the angle of rotation. For example, the five-pointed star can be rotated 72 degrees about its center, and it will look exactly as it did in its original position. An equilateral triangle, which has sides of equal length, can be rotated 120 degrees about its center and it will still look the same. Many objects in nature, such as some flowers and snowflakes, exhibit rotational symmetry. In the video segment, the flowers that exhibit rotational symmetry all had five petals, which means that their angle of rotation would be 360/5 degrees, or 72 degrees.
Plants and animals that generally exhibit symmetrical features are thought to be healthier than asymmetrical members of their species. Scientists believe that humans and animals inherently associate symmetrical features with a strong immune system, which is seen as a likely predictor of strong and healthy offspring. Researchers have found that animals and humans often seek out mates with symmetrical features.
In addition to nature, symmetry exists in many man-made objects and is an important concept in art, science, and architecture. When choosing their brand logos, companies have taken advantage of the fact that the human eye is attracted to symmetry. If you think about some of the sports team and company logos you know, you may recognize that many of them exhibit bilateral or rotational symmetry.
To learn about the symmetry found in snowflakes, check out Snowflake Physics Flash Interactive.
Bianca: I'd give anything for a green thumb. No matter how hard I try I kill every plant I own... it's a pattern, one that I need to break. You've heard about homicide? I'm guilty of herbicide! I water my plants every day. There's something wrong. I need help from an expert. (It's okay sweeties, we're gonna get you some water).
This is amazing; so many gorgeous plants! And none of them are dead. Can you help me? I have an emergency.
Gwen: Should I call an ambulance?
Bianca: Well, not that kind of emergency. I'm a plant killer, and I want to change.
Gwen: What do you mean?
Question: Why are plants and animals with symmetrical features often considered healthier? Answer: Because symmetrical features are associated with a strong immune system, which is seen as a predictor of strong and healthy offspring
Question: Which of the following is NOT a type of symmetry mentioned in the text? A) Bilateral symmetry B) Radial symmetry C) Rotational symmetry Answer: B) Radial symmetry
Question: What is the line that divides a figure with bilateral symmetry called? Answer: Line of symmetry | 677.169 | 1 |
You could describe this problem with two vectors an and orientation. The position of the camera $(x,y,z)_C$. The direction that the camera is pointing and the distance to the focal point (this is another vector) $(x,y,z)_F$. If you fix the distance to the focal point then you could describe this in terms of two angles. A third angle would be the rotation of the camera. This you need to define in terms of some reference (such as up - and a projection is involved). Transformations on the three angles won't commute but smooth arcs can be designed in that space. – AliceSep 8 '11 at 12:59
Question: Can the transformations on the three angles commute? Answer: No, they do not commute. | 677.169 | 1 |
Right Angles Geometry
Get more like this in a workbook
About This Worksheet
A right angle is an angle that's exactly 90 degrees. Help your child learn about right angle geometry with this simple geometry worksheet. She'll be asked to look at an angle and label it as either "less than" or "more than" a right angle. After completing this worksheet, she'll be better prepared for identifying different kinds of angles, a skill that's essential as she begins geometry.
Question: What is the main focus of this worksheet? (a) Practicing addition (b) Learning about right angles (c) Solving word problems Answer: (b) Learning about right angles | 677.169 | 1 |
In this lesson, let's learn how we classify triangles. And we will identify each of the triangles given below as either equilateral isosceles, or scalene which are basically a measure of whether this sides are equal or not. And then identify each of them as acute, right obtuse which is a measure of their angles.
So every triangle is named depending on whether the sides are equal or not and whether the angles are what size are the angles. So here's a simple rule to follow. If all three sides are equal, it's an equilateral triangle. If two sides are equal, it's an isosceles triangle. If no sides are equal, then it's a scalene triangle. On the angle side, here's the rule of thumb. If all three angles are equal, well that's called an equilateral triangle anyway. But if one of the angles equals 90 degrees, then it's a right triangle. If one of the angles is greater than 90degrees, then it is an obtuse triangle. If all angles are less than 90 degrees, then it's an acute triangle. We don't have to worry about all three angles being equal because all three angles being equal will give us all three angles being 60 degrees which is acute anyway.
So now let's change the color of our pen and look at each one of this, what can we see? Well all three sides are different sizes. So immediately I can say this one is a scalene triangle. And what's the size of the angles less than 90, less than 90, less than 90, so this is an acute scalene triangle. Here two sides are equal, two sides equal, isosceles. Angles all smaller than 90 degrees, so acute isosceles triangle. What do we see here all three sides are different, so it's scalene all three sides different scalene. And we see one angle is 90 degrees so it is a right scalene triangle.
Let's scroll on and do the second set. What do we see here? We see all three sides are equal the moment all the three sides are equal that's an equilateral triangle. If the equilateral triangle all angles are 60 degrees, so it makes an acute anyway. Here we see two sides are equal right, so it's an isosceles triangle. But we see that the angle is greater than 90 degrees, so this is an obtuse isosceles triangle.
And third one we see the angle as right 90 degrees, so it's a right triangle and two sides are equal so that's a nice isosceles triangle, right? Key thing to remember is we name a triangle depending on first whether the angles are one angle is right which is 90 degrees, it's greater than 90 or all three angles less than 90. And the second part of the naming is whether the sides are all equal and which case it's an equilateral to or equal isosceles and I'm not on the sides are the same scalene
Question: What is the name of a triangle with one angle measuring 90 degrees? Answer: Right triangle
Question: What is the first step in identifying a triangle, according to the text? Answer: Determine whether the sides are equal or not
Question: What is the name of a triangle with no sides of equal length? Answer: Scalene triangle | 677.169 | 1 |
Angles/276147: Three angles of a pentagon are 130', 90', and 80'. Of the remaining two angles, one is 30' more than twice the other. What is the sum of the smallest two angles? 1 solutions Answer 201300 by solver91311(16877) on 2010-03-02 13:28:23 (Show Source):
I'm going to just assume that you used ' to mean degrees of arc as opposed to minutes of arc which is the traditional meaning of that symbol.
The sum of the interior angles of any -gon is given by:
For a pentagon, is 5. So calculate the sum of the interior angles of a pentagon.
From that calculation result, subtract the sum of 130°, 90°, and 80° leaving you the sum of the measures of the remaining two angles.
Let represent the measure of the smaller angle. Then the larger angle must measure . Furthermore, the sum:
must be equal to the result of the previous calculation. Putting it all together:
Just solve for to get the measure of the smaller of the two remaining angles. This will turn out to be the smallest of all 5 angles. The other one will be the largest of all 5. Discover its measure by multiplying the measure of the smallest angle by 2 and adding 30 degrees. Now that you know the measure of all 5 angles, it should be an easy matter to add the measures of the two smallest ones.
John
Percentage-and-ratio-word-problems/276143299 by solver91311(16877) on 2010-03-02 13:12:32 (Show Source):
I just solved this one yesterday. The only thing I didn't do was express the probability as a percent -- but if you can't change back and forth between fractions and percentages by now, you are in way over your head. Go back a retake pre-Algebra.
You can work a calculator as well as I can. The others are worked exactly the same way, just change the value for .
John
Probability-and-statistics/276139: Bob is trying to borrow $1000 from friends. One of every 3 friends asked responds that their money is tied up. Of his friends with money not tied up, 2 out of 5 inform him quickly that they won't lend him money. Of the remainder, 1 out 2 won't lend him money on general principles. If Bob called 60 friends, how many might lend him money? 1 solutions Answer 201295 by solver91311(16877) on 2010-03-02 13:00:15 (Show Source):
Question: Out of the friends with money not tied up, what fraction informs Bob quickly that they won't lend him money? Answer: 2/5 | 677.169 | 1 |
Of course, the sides of this triangle satisfy the Pythagorean Theorem
but one reason I like this particular right triangle so much is the role it plays in another favorite triangle. The 5-12-13 triangle fits together perfectly with the 9-12-15 right triangle
to make the 13-14-15 triangle!
The 13-14-15 triangle is special in its own right: it is a Heronian triangle, a triangle with rational side lengths and rational area. In fact, this triangle has integer side lengths and integer area, making it especially interesting!
Happy Right Triangle Day! Be sure to marvel at some perpendicularity today.
Preliminary applications for the 2013 Rosenthal Prize for Innovation in Math Teaching are due Friday, May 10th.
The Rosenthal Prize, presented by the Museum of Mathematics, is designed to celebrate and promote the development and sharing of creative, engaging, replicable math lessons. The author of the winning activity receives $25,000, and the lesson will be freely shared with teachers around the world by the Museum of Mathematics.
Although May 10th is fast approaching, this first deadline is just a preliminary one. If the application process is anything like last year's, all that is required at this early stage are a few short essays about teaching philosophy and the overviews of the lessons you intend to submit in the fall. If an applicant passes through the preliminary stage, a more comprehensive application portfolio will likely be due in the fall of 2013. Again, this assumes the process is similar to last year's.
If you've got some fun, engaging, and replicable math lessons to share with the world, consider applying for the Rosenthal Prize! More information can be found here.
As runner-up for the Rosenthal Prize, I was interviewed for the piece, and had a chance to talk about my teaching philosophy, my award-winning lesson, and the value of hands-on, collaborative activities in mathematics class. In summarizing my approach to teaching mathematics, I said
"I want the classroom to be a place where we explore ideas together, where students can play around, experiment, collaborate, argue, create, and reflect on everything."
The purpose of the Rosenthal Prize is to encourage and promote innovative, replicable math activities that engage and excite students. I'm honored to be a part of this endeavor, and I look forward to more fun and creative math lessons being shared in the future.
Question: Which theorem does the triangle in the first sentence satisfy? Answer: The Pythagorean Theorem
Question: What is the author's teaching philosophy as a runner-up for the Rosenthal Prize? Answer: "I want the classroom to be a place where we explore ideas together, where students can play around, experiment, collaborate, argue, create, and reflect on everything." | 677.169 | 1 |
To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.
Crockett Johnson's painting directly imitates Descartes's figure found in Book I of La Géométrie. A translation of part of Book I is found in the artist's copy of James R. Newman's The World of Mathematics. The figure on page 250 is annotated.
This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.
Question: What is the series number of the painting? Answer: #36 | 677.169 | 1 |
In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length or magnitude and a direction to vectors. In turn, the notion of direction is strictly associated with the notion of an angle between two vectors. When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length (the square root of the dot product of a vector by itself) and angle (a function of the dot product between any two vectors). In three-dimensions, it is further possible to define a cross product which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of the parallelogram).
However, it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors).
Generalizations
In more general sorts of coordinate systems, rotations of a vector (and also of tensors) can be generalized and categorized to admit an analogous characterization by their covariance and contravariance under changes of coordinates.
In mathematics, a vector is considered more than a representation of a physical quantity. In general, a vector is any element of a vector space over some field, often represented as a coordinate vector. The vectors described in this article are a very special case of this general definition (they are not simply any element of Rd in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector.
Representation of a vector
Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices.) Other conventions include vec{a} or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as overrightarrow{AB} or AB. The hat symbol (^) is typically used to denote unit vectors (vectors with unit length), as in boldsymbol{hat{a}}.
Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here the point A is called the origin, tail, base, or initial point; point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector's magnitude, while the direction in which the arrow points indicates the vector's direction.
Question: How are vectors typically denoted? Answer: Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a.
Question: What is the general definition of a vector in mathematics? Answer: In mathematics, a vector is any element of a vector space over some field.
Question: What does the dot product characterize besides length? Answer: The dot product also characterizes the angle between two vectors. | 677.169 | 1 |
7th grade introduction/conclusion id had lost my 4square plan and i had wrote my introduction on the same paper and im out of ideas and its due monday i need help fast
geo (finals) please help! i dont get these problems at all! a(4,3) b(7,4) and c(5,2) are midpoints of a triangle. find the coordinates for the vertices of the triangle!
Geometry determine all possible values of x if 2x+10, 20-x, and 3(x-15) are the lengths for sides of a triangle
geometry conditional form I have a lot of cavities if I like candy a. Write the conditional form b. write the converse of the given conditional
Geometry Given tri ABC where A(4,-6) B, (-8,2) C (-4,8) A. Write the equation of the altitudes of triangle AC b. Determine the point of concurrency of the altitudes c. What is this point of concurrency called?
Question: What is the condition given in the conditional form problem? Answer: The condition is "I have a lot of cavities" | 677.169 | 1 |
But EF equals EB, therefore the rectangle CF by FA together with the square on AE equals the square on EB.
But the sum of the squares on BA and AE equals the square on EB, for the angle at A is right, therefore the rectangle CF by FA together with the square on AE equals the sum of the squares on BA and AE.
Subtract the square on AE from each. Therefore the remaining rectangle CF by FA equals the square on AB.
Now the rectangle CF by FA is FK, for AF equals FG, and the square on AB is AD, therefore FK equals AD.
Subtract AK from each. Therefore FH, which remains, equals HD.
And HD is the rectangle AB by BH, for AB equals BD, and FH is the square on AH, therefore the rectangle AB by BH equals the square on HA.
Therefore the given straight line AB has been cut at H so that the rectangle AB by BH equals the square on HA.
Q.E.F.
This construction cuts a line into two parts to solve the equation a (a – x) = x2 geometrically.
This construction is used in the proof of IV.10, which is later used to construct a regular pentagon. It accomplishes the same thing as the construction of proposition VI.30, which cuts a line into extreme and mean ratio, defined in VI.Def.3, and that construction is used
later in XIII.17.
The difference between this proposition and VI.30 is a matter of terminology. Propositions dealing with ratios of lines are postponed until Book VI, but any ratio concerning lines can be converted into a statement about areas of rectangles. Proposition VI.16 states that the line A is to the line B as the line C is to the line D is equivalent to the statement that the rectangle A by D equals the rectangle B by C.
The construction of this proposition cuts a line into two parts A and B so that the rectangle A + B by A equals the square B by B. The construction in VI.30 cuts a line so that A + B : B = B : A, which by VI.16, or by its special case VI.17, is the same thing.
Construction steps
For the purposes of cutting the line AB, the entire diagram does not have to be constructed. The points D, G, and K are unnecessary. In the diagram to the right, only those lines and circles necessary for the construction are shown, and only those parts of them that are relevant.
Question: What is the difference between this proposition and VI.30? Answer: The terminology used; they accomplish the same thing but with different ratios. | 677.169 | 1 |
The Shape Show- very misleading, wrong maths.. 2D shapes cannot be picked up. However "thin" the shape is,if it can be picked up, it is 3D. There was no mention of right angled triangles in the list of triangles & no differentiation between a square and an oblong. "Rectangle" was used to describe an oblong, but not a square. This resource should be removed. There are too many children who think that thin cuboids are rectangles.
Question: What is the issue with children learning from this resource regarding thin cuboids? Answer: Children may incorrectly think that thin cuboids are rectangles. | 677.169 | 1 |
When dealing with lines and points, it is very important to be
able to find out how long a line segment is or to find a
midpoint. However, since the midpoint and distance
formulas are covered in most geometry courses, you can
click here
to better your understanding of the midpoint and distance
formulas.
Circles, when graphed on the coordinate plane, have an equation
of x2 + y2 = r2 where
r is the radius (standard form) when the center of the
circle is the origin. When the center of the circle is
(h, k) and the radius is of length r, the equation
of a circle (standard form) is (x - h)2 + (y - k)2
= r2. Example:
1. Problem: Find the center and radius of
(x - 2)2 + (y + 3)2 = 16.
Then graph the circle.
Solution: Rewrite the equation in standard form.
(x - 2)2 + [y - (-3)]2 = 42
The center is (2, -3) and the radius is 4.
The graph is easy to draw, especially
if you use a compass.
The figure below is the graph of the solution.
Ellipses, or ovals, when centered at the origin, have an equation (standard form)
of (x2/a2) + (y2/b2) = 1.
When the center of the ellipse is at (h, k), the equation (in
standard form) is as follows:
(x - h)2 (y - k)2
-------- + -------- = 1
a2 b2
Example:
1. Problem: Graph x2 + 16y2 = 16.
Solution: Multiply both sides by 1/16
to put the equation in
standard form.
x2 y2-- + -- = 116 1a = 4 and b = 1. The
vertices are at (±4, 0) and
(0, ±1). (The points are
on the axes because the equation
tells us the center is at the origin,
so the vertices have to be on the
axes.)
Connect the vertices to form an oval,
and you are done!
The figure below is the graph of the ellipse.
The equation of a hyperbola (in standard form) centered at the
origin is as follows:
x2 y2
-- - -- = 1
a2 b2
Example:
1. Problem: Graph 9x2 - 16y2 = 144.
Solution: First, multiply each side of the
equation by 1/144 to put it
in standard form.
x2 y2-- - -- = 116 9
We now know that a = 4 and
b = 3. The vertices are at
(±4, 0). (Since we
know the center is at the origin,
Question: Which of the following is NOT a standard form equation for a circle? A) x² + y² = r² B) (x - h)² + (y - k)² = r² C) (x - h)² - (y - k)² = r² Answer: C) (x - h)² - (y - k)² = r².
Question: What is the center of the circle in the first example? Answer: The center is (2, -3). | 677.169 | 1 |
I IThe trick is that: Firstly you cut triangles from the one width and one length, if there's no overleft, there won't be any overleft.
In this case, 12 is divisible by 4 and 3
7= 3+4
For sure, there won't be any overleft when you cut from one width and one length.
In short, as long as length/width( one of the two) is sum of the triangle's two legs ANDwidth/length( one of the two) is the product of the triangle's two legs, there won't be overleft.
Question: What does the text mean by "overleft"? Answer: Leftover material after cutting | 677.169 | 1 |
: I'm enrolled in a online school and my text is online. The text is very vague in showing me how to do stuff. I am having trouble with a question. If anyone could help it would be much obliged. The question is:
: A right-angled triangle has vertices A(3,4), B(7,-4), C(-5,0). Show that the midpoint of the hypotenuse is equidistant from each vertex.
Ok, that sounds like a silly question. Isn't the midpoint of a line segment by definition the point that is equidistant from each end? But, let's humor them. You can draw some lines on the graph & then either use the Pythagorean theorem to calculate the length of each half. Or, you can draw some lines & then just argue the point logically.
Call point (1,0) D. Call point (1,-2) E. Call point (1,-4) F. Draw line FB from (1,-4) to (7,-4). Draw line DF from (1,0) to (1,-4). Line BF (part of the line y = -4) is obviously parallel to the X axis, so angle DCE is equal to angle EBF (alternate interior angles theorem). Also, angle CDE equals angle EFB (same reason, or, they are both right angles since line x=1 is perpendicular to lines y = 0 and y = -4). Line CD (6 units along x axis from (-5,0) to (1,0) equals line FB (6 units along line y=-4 from (1,-4) to (7,-4). So, by angle-side-angle, triangles CDE and BFE are congruent. Therefore, hypotenuse CE must equal hypotenuse EB.
Question: What is the length of line CD? Answer: 6 units
Question: Which theorem is used to prove that triangles CDE and BFE are congruent? Answer: Angle-Side-Angle (ASA) Congruence Theorem | 677.169 | 1 |
8.3.1 Connect the measurement of length, liquid, and mass with the appropriatecustomary and Metric units.
8.3.2 Convert units within and between measurement systems using conversionratios. Convert between various units of area and various units of volume. Calculate and convert rates through conversion ratios and label correctly.
8.4.1 Describe, classify, and compare geometric figures and solids.
8.4.2 Find the missing sides of similar figures using proportions. Find the missing sides of right triangles using the Pythagorean Theorem.
12.4.3 Classify and compare two- and three-dimensional shapes in terms of congruence and similarity and apply the relationships. Apply properties, draw images, and make conclusions of and about transformations (translations, reflections, rotations, and glides).
12.4.4 Calculate distance and midpoints between two points on a coordinate graph. Draw right triangles between any two points and apply the Pythagorean Theorem.
12.4.5 Know and use the definitions of sine, cosine, and tangent. Estimate values of trigonometric ratios. Use sin, cos, and tan to determine unknown lengths in real situations.
12.4.6 Find missing angles, arcs, chords made by the intersection of secants, tangents and circles. Determine measurements of angles formed by parallel, perpendicular and transversal lines. Use the Polygonal Sum Theory and the Pythagorean Theorem.
12.4.4 Transform an equation of a conic section (general form to/from vertex form) and graph.
12.4.5 Approximate and give exact values of trigonometric functions with and without a calculator. Determine the measure of an angle given its sine, cosine, or tangent. Identify and use definitions and theorems relating sines, cosines, and tangents. Solve real-world problems and find missing parts of triangles using right triangle trigonometry, Law of Cosines, and Law of Sines.
12.5.1 Use matrices to store data and to represent and solve real-world situations.
12.5.2 Fit an appropriate model to data (linear, quadratic, power, exponential). Relate the slope (rate of change) and intercepts (initial values) of a regression line. Predict values from a mathematical model (linear, quadratic, power, exponential). Consider the correlation coefficient and determine the validity of predictions made from regression equations.
12.5.4 Identify relationships between figures and their transformation images.
12.6.3 Write, solve and graph linear inequalities in one and two variables.
12.6.4 Identify patterns as sequences (geometric or arithmetic) and write general and recursive definitions. Identify, translate, write, graph and solve variation problems. Determine whether a relation defined by a table, a list of ordered pairs or a simple equation is a function.
12.... Transform equations from vertex form to standard form, and vise versa. Use the discriminant of a quadratic equation to determine the nature of the solutions to the equation.
Question: What is the name of the algebraic structure used to represent relationships between figures and their transformation images? Answer: Transformation matrices.
Question: In which section is the concept of transforming an equation of a conic section discussed? Answer: 12.4.4.
Question: Which mathematical concept is used to convert units within and between measurement systems? Answer: Conversion ratios. | 677.169 | 1 |
Question 252434: triangle A is similar to triangle B. side length of triangle A is 3 inches .side length of triangle B is 12 inches. the area of A is 10. what is the area of B ?
a 20 b 40 c 160 d 640 e 40640 Click here to see answer by drk(1908)
Question 253146: Tim says that if the measure of one angle of an isosceles triangle equals 30 degrees, then the measure of the other two angles must each be 75. Marc disagrees he says that while Tim is correct some of the time, there is another possible solution.support Tim's claim and Marc's claim.explain how the angle measures where found please. Click here to see answer by edjones(7569)
Question 253407: i have two questions that i really need help with, help would be very appreciated.
1. A parcel of land is in the shape of an isosceles triangle. the base had a length of 673 feet and the two equal legs meet an angle of 43 degrees. Find, to the nearest square foot, the area of the parcel of land.
i know that area equals one half base times height but i have no idea how to figure out the height i dont know if i should cut the triangle in half or what.
2. The triangle top of a table has two sides of 14 inches and 16 inches and the angle between the sides is 30 degrees. Find the area of the tabletop, in square inches.
i drew i diagram and have attepmted several ways but none of the answers makes sense should i use sohcahtoa i desperatley need help. Click here to see answer by Alan3354(30993 stanbon(57347 scott8148(6628)
Question 250244: I don't have scientific calculator, I need solution for this problem. :-
The measures of the angles of a triangle are (9 square root of 2x + 17 to the power 0) , (9 squareroot x) to the power 0 and (12 squareroot of x + 33 ) to the power 0 . find the measure of each angle? classify the triangle by its angles ? Click here to see answer by richwmiller(9135)
Question: What are the measures of the angles of a triangle with angle measures of (9√2x + 17)^0, (9√x)^0, and (12√x + 33)^0? Answer: The angle measures are 90 degrees, 90 degrees, and 0 degrees.
Question: What type of triangle is formed by the angle measures of (9√2x + 17)^0, (9√x)^0, and (12√x + 33)^0? Answer: The triangle is a right-angled triangle. | 677.169 | 1 |
Every week over 3000 museum visitors use the Triangle Tiling exhibit developed by the Geometry Center in collaboration with the Science Museum of Minnesota. The program features mathematical concepts such as the relationship between Platonic and Archimedean solids, and the dual of a polyhedron. The program is also used extensively at the Center itself during interactive tours, and will be on display at the SIGGRAPH computer graphics conference in July 1994. This picture is a collage of the 5 Platonic solids and 11 of the 13 Archimedean solids. The names, starting at the top left, are: dodecahedron, truncated dodecahedron, icosidodecahedron, truncated icosahedron, icosahedron, rhombicosidodecahedron, rhombitruncated icosidodecahedron, cube, truncated cube, cuboctahedron, truncated octahedron, octahedron, rhombicuboctahedron, rhombitruncated cuboctahedron, tetrahedron, truncated tetrahedron.
How to make it: Triangle Tiling is an external module of Geomview that runs on SGI workstations.
Question: Which of the following is NOT a mathematical concept featured in the Triangle Tiling exhibit? A) Pythagorean theorem B) Dual of a polyhedron C) Golden ratio Answer: A) Pythagorean theorem | 677.169 | 1 |
No. Yes. Yes. No. No. Only a rhombus can be divided into two equilateral triangles, and a rhombus is
a parallelogram. A rhombus with two 60 degree angles and two 120 degree angles can be split by one of its
diagonals (the one that bisects the 120 degree angles) into two congruent, equilateral triangles.
Question: Can a square be divided into two equilateral triangles? Answer: No. | 677.169 | 1 |
Human Conics
In this lesson students use sidewalk chalk and rope to illustrate the locus definitions of ellipses and parabolas. Kinesthetics, teamwork, and problem solving are stressed as students take on the role of focus, directrix, and point on the conic, and figure out how to construct the shape.
Learning Objectives
Students will:
Define conic sections as a locus of points
Apply locus definitions to draw conic sections
Collaborate with partners to solve a problem
Materials
Sidewalk chalk
Lightweight rope (about 10-12 feet per group)
Right angle measures (e.g., 8.5×11 sheets of cardstock or right angle rulers)
Instructional Plan
This lesson presents analogous approaches for the locus definition of three conics, the circle, the ellipse, and the parabola. The circle may be too easy for some students and the parabola may be too difficult for some. The overall lesson is divided into two parts, classroom and outdoor. Plan the conics you will use ahead of time so you can do the classroom sections together before going outdoors to do the associated activities. Also in preparation for the lesson, mark the midpoint of each rope with a permanent marker or a piece of tape.
For a large group demonstration, replace the pencil in a compass with an overhead marker and demonstrate the use of the compass on a transparency. Emphasize the importance of not squeezing the compass so that the radius is maintained. Let students practice by having each of them draw a circle with a radius equal to the length of their index finger. Discuss how the construction is related to the locus definition. Remind students that the circle is just the locus of points, not its interior.
Show students the Ellipse Definition overhead, covering the title, and ask what shape they see. If students say oval, explain that an oval and an ellipse may look alike, but the shapes we deal with in our study of conics are called ellipses. Indicate the foci and simply state that these are called focal points or foci. Give students the definition of an ellipse. Point out that "foci" is the plural of focus. Use different colors to illustrate the definition by selecting points on the ellipse and drawing lines to the foci. Do not answer questions or engage students in discussion so that students may ponder the definition as they work outside.
Show students the Parabola Definition overhead, covering the title, and ask what shape they see. If students have previously worked with parabolas, explain that considering the graph of a quadratic function is only one way of looking at it. Give students the locus definition. Remind students that the distance from a point to a line is the perpendicular to that line. Use different colors to illustrate this for several other points on the parabola.
Question: Which conic section might be too difficult for some students? Answer: The parabola. | 677.169 | 1 |
Before going outside, separate students into groups of three. Three students are needed for the ellipse and parabola, two to represent foci or directrix and one to draw. For the circle, only two students are required, the center and the chalk, but it is usually easier not to rearrange groups mid-activity. Explain to students that they will be working in groups to draw a circle, an ellipse, and a parabola. Each group will have one piece of chalk and one piece of rope. Do not give instructions or hints on how to draw the conics until students have had ample opportunity to experiment.
Circle
Instruct students to draw a perfect circle using the chalk and the rope. If students need a hint, suggest that they consider themselves to be a human compass.
If students need further instruction: Fold the rope in half. One student puts the ends together, and holds them on the ground to be the center of the circle. The second student stretches the rope and puts the chalk in the bend at the midpoint. The second student then drags the chalk along the ground, while pulling the rope taut. Note that students figuring out the activity independently may not fold the rope. This is not a problem.
As you observe students, ask how many students were actually needed to draw the circle? [two] What were their roles? [center of the circle and point on the circle] What did the rope represent? [the constant distance or radius]
Ellipse
If students need hints, tell them that the fact that there are three people in the group is significant and to consider what they did to draw the circle.
If students need further instruction: Two students are human foci, holding the ends of the rope at fixed points on the ground. These students should not hold the rope taut. The third student uses the chalk to pull the rope taut and sweeps out the locus of points.
As students finish, ask them to consider and discuss the questions on the activity sheet.
Parabola
Groups who finish the ellipse should begin experimenting with the parabola. If possible, suggest that students use an existing straight line such as a parking space or sidewalk crack as the directrix. Some students may need to see a demonstration before effectively drawing the parabola. Gather those students needing an outdoor demonstration. Have one group of students demonstrate the parabola drawing as you direct them through the instructions:
Draw a focus approximately 2 feet from the directrix. This does not need to be precise. You are just looking for a distance that will allow students room to maneuver and will produce an easily recognizable parabola.
Assign roles to the three students in the group: F, D, and A. Student D will be responsible for the directrix and will need a right angle measure, such as cardstock or a right angle ruler, to approximate right angles. Student F will be responsible for the focus of the parabola. Student A will mark points on the parabola.
Assign each student a point on the rope. Student A is at the marked midpoint of the rope. At equal distances from her, measured by folding the rope, are F and D.
Question: What should the two students playing as human foci not do? Answer: They should not hold the rope taut
Question: What is the main difference between drawing a circle and an ellipse? Answer: The number of students needed (two for a circle, three for an ellipse) and the fact that in an ellipse, the rope is not held taut by the foci
Question: How many students are needed to draw a perfect circle? Answer: Two
Question: What does the rope represent when drawing a circle? Answer: The constant distance or radius | 677.169 | 1 |
This is the basic interface element in both TrigAid and Physics 101 SE. This is called a formula box, it allows the calculation of the main formula, but also its inner variables. Clicking the calculate button yields the main calculation (editfield in gray) and clicking the radial buttons yields the subcalculations. Clicking the "?" shows the formula in either a popup window or below.
The geometry section relies on the formula box for its user interface. Simply enter in your given information into the editfields and click calculate. You may then click the "?" button to see the formula used and a numerical representation of the equation. In addition, clicking the radial buttons (round interface elements) gives the calculatin of that variable when the other fields are supplied with data.
The Triangulator is a powerful feature, it will try to find based upon the information given to it. First ascertain the type of triangle it is and select it in the popupmenu. If any of the sides is given by an exact value, click the appropiate checkbox to use root notation. Next enter in as much information as you can about the sides and angles and click calculate. The program will try to fill in as much information as it can.
This is a straightforward section. In the first two altered formula boxes is the law of sines and cosines. You must select through a popupmenu the side or angle you are solving for, enter in the required information and click calculate.
The Famous SOHCAHTOA is set up according to the foundation of its saying so it can be easily used. Just enter in the information and click on the radial button of the unknown quantity.
Lastly there is Pythagoren's Theorem, enter in two sides and click the radial button of the desired unknown quantity. Finally, you can find the formula by clicking "?" or the buttons in the SOHCAHTOA groupbox.
To find the zero of a polynomial enter in the coefficient into the equation above, put a zero in a term that does not exist. Click the "Insert Coefficients" button and uses the up and down arrows to find a zero of that polynomial. A zero is found when the last position is zero. When this is found click "Shift and Store" and repeat the process as needed.
The first formula box converts a log based expression into one including an exponent. The second uses the change of base formula to calculate logs with uncommon bases. The third one solves for a exponent and creates a logarithm and finally the last solves exponential growth and decay problems.
Enter in the magnitude (a length, or velocity, etc) and their corresponding angle in degrees. This will yield a vertical magnitude, horizontal magnitude, overall magnitude and angle. It is important that the magnitudes and angles be entered in the correct pairs, or they will mismatch and produce incorrect answers. The reference angle is there to provide a reference where the angle can be based off of.
Question: What does the SOHCAHTOA groupbox provide in the last section? Answer: It provides the formula for the trigonometric functions sine, cosine, and tangent.
Question: What is the purpose of the 'Insert Coefficients' button in the polynomial zero-finding section? Answer: To insert the coefficients of the polynomial into the equation.
Question: What is the first step in using the Triangulator feature? Answer: First, ascertain the type of triangle it is and select it in the popupmenu. | 677.169 | 1 |
Definitions
Century Dictionary and Cyclopedia
n. In anc. math., a line compounded of two medials. If these latter make a rational rectangle, the compound is called a first bimedial; if they make a medial rectaugle, the compound is termed a second bimedial. In modern language this would be expressed by saying that a bimedial is a quantity of the form (√a + √b) √c, where a, b, and c are commensurable. It is a first or a second bimedial according as a b c is or is not a perfect square.
Question: True or False: A bimedial is always a rational number. Answer: False. | 677.169 | 1 |
==> geometry/konigsberg.p <== Can you draw a line through each edge on the diagram below without crossing any edge twice and without lifting your pencil from the paper?
+---+---+---+ | | | | +---+-+-+---+ | | | +-----+-----+
==> geometry/konigsberg.s <== This is solved in the same way as the famous "Seven Bridges of Konigsberg" problem first solved by Euler. In that problem, the task was to find a closed path that crossed each of the seven bridges of Konigsberg (now Kaliningrad, Russia) exactly once. For reasons given below, no such path existed. In this version, you cannot draw such a line without cheating by:
(1) drawing a line along one of the edges, or (2) inscribing the diagram on a torus, or (3) defining a line segment as the entire length of each straight line, or (4) adding a vertex on one of the line segemnts, or (5) defining "crossing" as touching the endpoint of a segment.
The method for determining if paths exist in all similar problems is given below.
Turn each "room" into a point. Turn each line segment into a line connecting the two points representing the rooms it abuts. You should be able to see that drawing one continuous line across all segments in your drawing is equivalent to traversing all the edges in the resulting graph. Euler's Theorem states that for a graph to be traversable, the number of vertices with an odd number of edges proceeding from them must be either zero or two. For this graph, that number is four, and it cannot be traversed.
To prove Euler's Theorem, think of walking along the graph from vertex to vertex. Each vertex must be entered as many times as it is exited, except for where you start and where you end. So, each vertex must have an even number of edges, except possibly for two vertices. And if there are two vertices with an odd number of edges, the path must start at one and end at the other.
==> geometry/ladders.p <== Two ladders form a rough X in an alley. The ladders are 11 and 13 meters long and they cross 4 meters off the ground. How wide is the alley?
==> geometry/ladders.s <== Ladders 1 and 2, denoted L1 and L2, respectively, will rest along two walls (taken to be perpendicular to the ground), and they will intersect at a point O = (a,s), a height s from the ground. Find the largest s such that this is possible. Then find the width of the alley, w = a+b, in terms of L1, L2, and s. This diagram is not to scale.
Now, (**) defines a fourth degree polynomial in y. It can be written in the form (by simply expanding (**))
(***) y^4 - 2s_y^3 - L_y^2 + 2sL_y - Ls^2 = 0
Question: Can a line be drawn through each edge of the given diagram without crossing any edge twice and without lifting the pencil? Answer: No, it is not possible. | 677.169 | 1 |
L1 and L2 are given, and so L is a constant. How large can s be? Given L, the value s=k is possible if and only if there exists a real solution, y', to (***), such that 2k <= y' < L2. Now that s has been chosen, L and s are constants, and (***) gives the desired value of y. (Make sure to choose the value satisfying 2s <= y' < L2. If the value of s is "admissible" (i.e., feasible), then there will exist exactly one such solution.) Now, w = sqrt(L2^2 - y^2), so this concludes the solution.
L1 = 11, L2 = 13, s = 4. L = 13^2-11^2 = 48, so (***) becomes
y^4 - 8_y^3 - 48_y^2 + 384_y - 768 = 0
Numerically find root y ~= 9.70940555, which yields w ~= 8.644504.
==> geometry/lattice/area.p <== Prove that the area of a triangle formed by three lattice points is integer/2.
==> geometry/lattice/area.s <== The formula for the area is
A = | x1*y2 + x2*y3 + x3*y1 - x1*y3 - x2*y1 - x3*y2 | / 2
If the xi and yi are integers, A is of the form (integer/2)
==> geometry/lattice/equilateral.p <== Can an equlateral triangle have vertices at integer lattice points?
==> geometry/lattice/equilateral.s <== No.
Suppose 2 of the vertices are (a,b) and (c,d), where a,b,c,d are integers. Then the 3rd vertex lies on the line defined by
(x,y) = 1/2 (a+c,b+d) + t ((d-b)/(c-a),-1) (t any real number)
and since the triangle is equilateral, we must have
||t ((d-b)/(c-a),-1)|| = sqrt(3)/2 ||(c,d)-(a,b)||
which yields t = +/- sqrt(3)/2 (c-a). Thus the 3rd vertex is
1/2 (a+c,b+d) +/- sqrt(3)/2 (d-b,a-c)
which must be irrational in at least one coordinate.
==> geometry/manhole.cover.p <== Why is a manhole cover round?
Question: What is the formula for the area of a triangle formed by three lattice points? Answer: A = |x1y2 + x2y3 + x3y1 - x1y3 - x2y1 - x3y2| / 2
Question: What is the value of L in the given context? Answer: L = 13^2 - 11^2 = 48
Question: What is the value of w calculated from the given y? Answer: w ≈ 8.644504 | 677.169 | 1 |
==> geometry/manhole.cover.s <== It will not fall into the hole, even if rotated, tipped, etc. It gives maximal area for a given amount of material. It does not have to be carried, but can be rolled. Human beings are roughly round in horizontal cross section. Orientation of the cover with the access hole is not of concern. Orientation of the access hole with the ladder in the pipe below is not of concern.
-- +------------------- [email protected] -------------------+ | The effort to understand the universe is one of the very few things | | that lifts human life above the level of farce, and gives it some | | of the grace of tragedy - Steven Weinberg | +---------------------------------------------------------------------+
==> geometry/points.in.sphere.p <== What is the expected distance between two random points inside a sphere? Assume the points are uniformly and independently distributed.
==> geometry/points.in.sphere.s <== Use spherical polar coordinates, and w.l.o.g. choose the polar axis through one of the points. Now the distance between the two points is
sqrt ( r1^2 + r2^2 - 2 r1 r2 cos(theta))
and cos(theta) is (conveniently) uniformly distributed between -1 and +1, while r1 and r2 have densities 3 r1^2 d(r1) and 3 r2^2 d(r2). Split the total integral into two (equal) parts with r1 < r2 and r1 > r2, and it all comes down to integrating polynomials.
More generally, the expectation of the n'th power of the distance between the two points is
==> geometry/points.on.sphere.p <== What are the odds that n random points on a sphere lie in the same hemisphere?
==> geometry/points.on.sphere.s <== 1 - [1-(1/2)^(n-2)]^n
where n is the # of points on the sphere.
The question will become a lot easier if you restate it as the following:
What is the probability in finding at least one point such that all the other points on the sphere are on one side of the great circle going through this point.
When n=2, the probability= 1 , when n=infinity, it becomes 0.
In his Scientific American column which was titled "Curious Maps", Martin Gardner ponders the fact that most of the land mass of the Earth is in one hemisphere and refers to a paper which models continents by small circular caps. He gives the above result.
See "The Probability of Covering a Sphere With N Circular Caps" by E. N. Gilbert in Biometrika 52, 1965, p323.
==> geometry/revolutions.p <== A circle with radius 1 rolls without slipping once around a circle with radius 3. How many revolutions does the smaller circle make?
Question: What is the primary advantage of the manhole cover's shape? Answer: It gives maximal area for a given amount of material.
Question: Who is the author of the Scientific American column "Curious Maps"? Answer: Martin Gardner
Question: Will the manhole cover fall into the hole if rotated or tipped? Answer: No, it will not fall into the hole.
Question: What is the probability that n random points on a sphere lie in the same hemisphere? Answer: 1 - [1-(1/2)^(n-2)]^n | 677.169 | 1 |
==> geometry/table.in.corner.p <== Put a round table into a (perpendicular) corner so that the table top touches both walls and the feet are firmly on the ground. If there is a point on the perimeter of the table, in the quarter circle between the two points of contact, which is 10 cm from one wall and 5 cm from the other, what's the diameter of the table?
==> geometry/table.in.corner.s <== Consider the +X axis and the +Y axis to be the corner. The table has radius r which puts the center of the circle at (r,r) and makes the circle tangent to both axis. The equation of the circle (table's perimeter) is
(x-r)^2 + (y-r)^2 = r^2 .
This leads to
r^2 - 2(x+y) + x^2 + y^2 = 0
Using x = 10, y = 5 we get the solutions 25 and 5. The former is the radius of the table. It's diameter is 50 cm.
The latter number is the radius of a table that has a point which satisfies the conditions but is not on the quarter circle nearest the corner.
==> geometry/tetrahedron.p <== Suppose you have a sphere of radius R and you have four planes that are all tangent to the sphere such that they form an arbitrary tetrahedron (it can be irregular). What is the ratio of the surface area of the tetrahedron to its volume?
==> geometry/tetrahedron.s <== For each face of the tetrahedron, construct a new tetrahedron with that face as the base and the center of the sphere as the fourth vertex. Now the original tetrahedron is divided into four smaller ones, each of height R. The volume of a tetrahedron is Ah/3 where A is the area of the base and h the height; in this case h=R. Combine the four tetrahedra algebraically to find that the volume of the original tetrahedron is R/3 times its surface area.
==> geometry/tiling/count.1x2.s <== The number of ways to tile an MxN rectangle with 1x2 dominos is 2^(M*N/2) times the product of
{ cos^2(m*pi/(M+1)) + cos^2(n*pi/(N+1)) } ^ (1/4)
over all m,n in the range 0<m<M+1, 0<n<N+1.
[Exercises: 0) Why does this work for M*N odd? 1) When M<3 the count can be determined directly; check that it agrees with the above formula. 2) Prove directly this formula gives an integer for all M,N, and further show that if M=N it is a perfect square when 4|N and twice a square otherwise. ]
Question: What is the count of ways to tile a 2x2 rectangle with 1x2 dominos according to the formula? Answer: 2*(1/2) = 1 (since cos^2(π/3) + cos^2(π/3) = 1) | 677.169 | 1 |
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1Plane and Solid Figures
On the GED® Mathematics Test, students will apply their understanding of circles to solve problems.
They need to understand that a circle is a flat, closed figure in which every point is the same distance from the center.
They also need to be able to use the concepts of radius and diameter and the relationship between the two.
This lesson focuses on the properties of circles. A challenge of the GED Mathematics Test is that students need to recognize a circle when it is embedded in a realistic setting or when parts of a circle are included in an irregular figure (as in Lesson 5 of this module).
2Key Circle Concepts
A circle is a flat, closed figure with no angles or sides. A complete rotation around the center of a circle is 360 degrees (°).
Step 1 Review the definition of a circle. Find the figures that are flat and closed.
Figures (2) and (5) are flat and closed, so they could be circles. Figures (1) and (3) have three dimensions, so they are not flat, and figure (4) is open.
Step 2 Check if the center is the same distance from every point on the figure. The center of the figure in choice (5) is not the same distance from every point, but the distance from center of choice (2) is.
Figure (5) is not a circle, figure (2) is a circle.
Answer figure (2)
3Radius and Diameter
The radius is the distance from the center of a circle to any point on the circle.
The diameter is the distance across a circle through the center. The diameter is always twice the length of the radius.
Question What is the diameter of the wheel in the image on the left?
Step 1 Find the length of the radius since the diameter is always twice the length of the radius.
radius = 13 inches
Step 2 Multiply the radius by 2 to find the diameter.
2 × 13 = 26
Answer 26 inches5Skill Review
Perimeter, Area, and Volume: Properties of Circles Skill Review
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In this lesson you have learned about the properties of circlesCredits
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Screen 2:
Screen 3:
Screen 4:Properties of Circles is the second of seven self-paced lessons in the "Perimeter, Area, and Volume" section of KET's GED® Geometry Professional Development Online Course. This lesson focuses on the properties of circles and guides students in recognizing a circle when it is embedded in a realistic setting or when parts of a circle are included in an irregular figure.
This course is designed to help you review and build your skills and knowledge of geometry concepts and to help you to gain confidence in preparing your learners for a substantial portion of the GED Mathematics Test. Click on the view button on the left to begin Lesson 2
Question: Which of the following is NOT a circle? (1), (2), (3), (4), (5)
Answer: (5)
Question: What is the next lesson in the "Perimeter, Area, and Volume" section of KET's GED® Geometry Professional Development Online Course?
Answer: Lesson 3 | 677.169 | 1 |
geometric designs, such as the one below, constructed by a geometry
student.
You will find a step-by-step set of instructions for constructing
a cardioid below:
1) Construct a circle using a compass. Construct a horizontal line
through the center of the circle, using a ruler. Construct a line
through the center perpendicular to the horizontal line. Point P is
the point that will begin our envelope. Now bisect the right angle
formed by these two perpendiculars. Continue to construct angle
bisectors until your circle is divided into 16, 32 or 64 equal parts.
The more divisions on the circle, the more complex your design will
be.
2) Construct a circle using one of the division marks on the given
circle as the center, and point P as a point on the circle. Continue
this process to create more circles, passing through point P.
You can color your construction in many different ways, to create
beautiful designs!
Question: What is the first step in constructing the cardioid design? Answer: Construct a circle using a compass. | 677.169 | 1 |
Understanding early years environments and children's spaces This unit considers some of the different environments children encounter in their early years. It encourages you to develop your reflection of children's environments and provides opportunities for you to investigate and evaluate young children's experiences and your role in supporting them. First published Author(s): Creator not set
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We are going to look at some of the basics of trigonometry relating to right angle triangles. So the first question is, What is a right angle triangle?
It is a triangle in which one of the angles is 90°, which is commonly referred to as a right angle. The sum of the angles in any triangle is 180°. So if the other two angles are α and β as
Question: What is the total number of angles in a triangle? Answer: Three | 677.169 | 1 |
This explanation may be incomplete or incorrect. If you see a way to improve it, edit it! Thanks.
In this comic, Randall attempts to draw a five-pointed star, which is one of the toughest things to draw for whatever reason. The title text references the fact that a five-pointed star has all angles at 36 degrees what sums to 180, like a common triangle. This pentagram is the simplest regular star polygon in geometry. Also, it references that one of the symbols of Judaism is a six pointed star which is much easier to draw because it is composed of two equilateral triangles. (Citation Needed) 5 pointed stars may be easier for some because they can be drawn in one continuous motion.
Question: What is the sum of the angles in a five-pointed star? Answer: 180 degrees | 677.169 | 1 |
To square the fence line to the house, you'll mark off a right triangle extending from the foundation. Sink one stake for the triangle's corner where the first post will go, and a second one 3 feet away along the foundation. Tie a mason line to the first stake, stretch it taut roughly perpendicular to the house, and mark it 4 feet from the stake. Find the 5-foot mark on a measuring tape and angle it from the second stake toward the line. Now cross the taut line and the tape until you get the 4-foot and 5-foot marks to meet. When they do, according to the Pythagorean theorem, you have a 90-degree angle at the triangle's corner—and thus a perpendicular line intersecting the house. A bigger triangle (say 9, 12, and 15 feet) works even better.
Question: What is the purpose of the measuring tape in this process? Answer: To create a 90-degree angle by intersecting the taut mason line and the tape at the 4-foot and 5-foot marks, respectively. | 677.169 | 1 |
from Chapter 3
Good Evening Everyone! It is time for us to bid farewell to Chapter 3 considering that we took our test today and are now moving on to the next chapter. Personally, I enjoyed chapter 3 because learning all of the postulates and theorems helped me to gain a new understanding on angles and corresponding parts, as well as polygons. One of my favorite problems from chapter 3 was this; Triangle RGT is a right triangle. Angle G is a right angle and angle R=19 degrees. Find the measurement of angle T.
It takes a few steps to solve this problem. First of all, you have to remember that a right triangle is equal to 90 degrees due to the Definition of a Right Triangle. Next you have to remember that the sum of all the angles in a triangle will add up to 180 degrees due to the Triangle Angle-Sum Theorem. Therefore, if you add the two measurements of the angles that you have together and subtract them from 180, then you will get the measurement of the missing angle. For example, 90+19=109, and 180-109=71. Therefore, the measurement of angle T is equal to 71 degrees.
I enjoyed this problem because it was similar to a puzzle that you have to solve in order to get the answer. It also relates to triangles, and I found that I really enjoyed working with triangles during this chapter! What was your favorite problem?
Question: If you know the measures of two angles in a triangle, what do you need to do to find the third angle? Answer: Add the two known angles together, then subtract that sum from 180 degrees. | 677.169 | 1 |
trig 53. A 60-foot drawbridge is 24 feet above water level when closed. When opened the bridge makes an angle of 33 degrees with the horizontal. a. How high is the tip P of the open bridge above the water? b. When the bridge is open, what is distance from P to Q? 53. A 60-foot ...
Friday, February 17, 2012 at 5:58am
Question: At what angle does the bridge open with respect to the horizontal? Answer: 33 degrees | 677.169 | 1 |
SOL
Algebra ->
Algebra
-> Triangles
-> SOLLog On
Question 144862ATION OF THE SUN IS 52 DEGREES, FIND THE HEIGHT OF THE BUILDING CORRECT TO THE NEAREST INTEGER.
***I MUST SHOW ALL WORK SO PLEASE DO SO FOR ME*** THANK YOU VERY MUCH***
THE PICTURE SHOWS RIGHT TRIANGLE WITH ONE 90 DEGREE ANGLE AND ONE 52 DEGREE ANGLE. I KNOW THE OTHER ANGLE WOULD BE 38 DEGREES. THE ONLY LENGTH I ASSUME GIVEN WOULD BE 23FT WHICH IS ONE LEG THE OTHER LEG IS NOT GIVEN NOR HYPOTENUSE.
THANK YOU SO MUCH IN ADVANCE! =] Answer by Earlsdon(6287) (Show Source):
You can put this solution on YOUR website! Ok, you can use the tangent function to solve this problem.
Remember that in a right triangle, such as you have here, the tangent of 52 degrees (tan(52) is the ratio of the side opposite the 52-degree angle (that's the height of the building, h) divided by the side adjacent to the 52-degree angle (that's the length of the building's shadow, or 23 ft.)
So, we can write: Now you multiply both sides by 23 to get: You can do this in your calculator. Round the the nearest integer: feet.
Question: True or False: The other angle in the triangle is 90 degrees. Answer: False
Question: What is the height of the building to the nearest integer, as calculated in the solution? Answer: 25 feet | 677.169 | 1 |
Ideas from Classroom Teachers for Trigonometry and Triangles
The applications section is the central point to this unit; the last few topics should also be addressed from an applications-motivated perspective. Note that anyone can select a formula and plug numbers in. The real challenge here is to create an appropriate diagram, determine how the given numbers and missing value match up to the variables in an appropriate formula, then substitute and solve (the easy part).
See Mathematics Teacher articles for good applications of this topic.
When working with Law of Sines and Law of Cosines, you will need to review basic properties of triangles such as the angle-sum (180°) and the triangle inequality. These are important since they help determine whether the output of the Law of Sines/Cosines has any validity.
If you choose to explore the ambiguous case: One difficult case conceptually is the ambiguous case of the Law of Sines, in which the given information (certain A-S-S combinations) may allow for 2 different triangle configurations. Allowing students to experience the ambiguity first, by creating triangles from spaghetti pieces, will make it seem more reasonable when the Law of Sines returns two possible angle measures.
Dynamic geometry software can be used to illustrate triangle similarity and trigonometric ratios.
Application topics could include angle of elevation (or depression), distance-rate-time, height of a mountain, etc.
Finding the area of non-right triangles can be considered in the context of proving the Law of Sines.
Students can use the formula for the area of any triangle to find the area of a regular pentagon (or other regular polygon).
A proof of Heron''s formula is accessible to students. Historical note: Although named for Heron (c. 1st century), Archimedes (c. 287-212 BC) proved it first.
Question: Which topics should be addressed from an applications-motivated perspective? Answer: The last few topics should be addressed from an applications-motivated perspective.
Question: What are the basic properties of triangles that should be reviewed when working with the Law of Sines and Law of Cosines? Answer: The basic properties of triangles that should be reviewed are the angle-sum (180°) and the triangle inequality. | 677.169 | 1 |
The proof is by outer five segment on D′ C R and P C E and points P and D′. We pretty much have everything we need to apply outer five segment, although between P C E is complicated enough to break out into its own lemma. This lemma follows from between P C C′ and between C E C′.
It might seem that having proved D′ = D we are done with this case (since after all, D′ = D is the long-term goal of this proof). Although we could, of course, proceed this way, it would entail carrying along the two cases to almost the end of the proof. So we instead will take a few more steps to prove C Q ≡ C D. The next one is P = Q.
We've set up the dominos, and in this section we push them over. For a domino (point) x to "fall" means that we prove x P ≡ x Q (via EquidistantLine). So we need three collinear points, the first two of which are not equal, and the first two of which already have their congruences with P and Q proven. That gives us the third congruence, and then this domino is ready to push on one of the following ones.
Question: What is the result of a domino (point) x falling in the text? Answer: It proves x P ≡ x Q | 677.169 | 1 |
Four Sided Polyg learn to identlfy polygons by determining whether they they are closed depending on if an animal can "escape" from inside the "fence." Students will also learn how to draw their own polygons.
NOTEBOOK (SMARTboard) File
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144.48
Question: What is the purpose of the text? Answer: To introduce a method for students to learn about identifying and drawing polygons. | 677.169 | 1 |
Question 461510: Consider a nine point circle(Feuerbach's circle, Euler's circle, Terquem's circle). How to prove that nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle. Answer by richard1234(5390) (Show Source):
Suppose we extend HF to a point X on the circumcircle of ABC, and denote the midpoint of HC with Y (this must also lie on the 9-point circle, by definition):
By the Power of a Point theorem, HF*HY = k for some constant k. In addition, since HC = 2HY and HX = 2HF, then applying Power of a Point theorem again, HC*HX = 4k.
Intuitively, it would seem that the 9-point circle bisects each segment from the orthocenter to the circumcircle. This is because it bisects all of our known segments (such as HX, HA, etc.) and what we have found from Power of a Point theorem supports this. However I haven't yet found a way to prove this holds for all points since I chose arbitrary points on the circles and couldn't prove that two triangles were similar with ratio 1:2. Can you show this works for all points? You could try using similarity, or some other technique such as projective geometry or extending BE, AD to points Y and Z on the circumcircle, then connecting X, Y, and Z to create another triangle with orthocenter H. I'm sure there are several ways to accomplish this.
Question: What is the name of the circle mentioned in the text? Answer: Nine-point circle
Question: What is the ratio of similarity between two triangles mentioned in the text? Answer: 1:2 | 677.169 | 1 |
Angle between a plane and horizontal
I need to find the angle between a set of planes ax+by+cz+d = 0 and horizontal plane (x-y plane).
For example my first plane equation is x - 4y +3z +1 = 0
so i wrote the following code
n1 = [1 -4 3]; % Normal vector to plane
n2= [0 0 1]; % normal vector to x-y plane
cosang = dot(n1,n2); % actually n1.n2 = |n1||n2|cosang
n1crossn2 = cross(n1,n2);
sinang = norm(n1crossn2); % actually n1*n2 = |n1||n2|sinang n
angle = atand(sinang, cosang);
it gives an answer 53.96
I thought the angle between plane and horizontal is same as the angle between normal to both plane. Is it true? I don't know how to verify this?
Also how can i check the sign of the angle(possitive or negative)?
Is there any other way to get the angle between given plane and horizontal plane?
Note: I don't know the coordinates of any plane lying on that plane. I only know the plane equation
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The angle between the two planes are the angle between the two normal vectors, so your approach is correct. I'm not sure why you need to do atand though. I think once you do the dot product, you can use acosd directily
Question: Why is the `atand` function not needed in this case? Answer: The `atand` function is used to find the angle from the tangent of the angle. However, in this case, we already have the cosine of the angle (from the dot product), so we can directly use `acosd` to find the angle.
Question: How can the sign of the angle (positive or negative) be determined? Answer: The sign of the angle can be determined by checking the cross product of the normal vectors. If the cross product points up (positive z-direction), the angle is positive. If it points down, the angle is negative. | 677.169 | 1 |
Trigonometric Graphs (page 1)
In this laboratory, we will examine trigonometric functions and their graphs.
Upon completion of the lab, you should be able to quickly sketch such functions
and determine such characteristics as period and amplitude. You should also
be able to determine whether the function has been shifted, reflected, stretched
or shrunk as compared to the graph of one of the six trigonometric functions
discussed in a previous laboratory.
Question: What are some characteristics of trigonometric functions that can be determined after completing this lab? Answer: Period and amplitude. | 677.169 | 1 |
You may have wondered, when we said that an array of
numbers was called a 'vector', why we used that term. Isn't a vector just
an arrow, or the line-based artwork that comes from programs like Illustrator
or Flash?
Well, one way to represent an array of numbers is as
a point on a grid. For instance, the vector [80,60] can be shown on a
2-dimensional grid:
More accurately, [80,60] describes the arrow, or vector,
from (0,0) to (80,60), shown here in white—that is, a vector which
goes 80 units across and 60 down. For an array of 3 numbers, we'd of course
draw a 3D vector:
So, why do we say that [80,60] represents the vector,
and not the point it leads to? Well, in part because the point is meaningless
when taken out of its context. One point is indistinguishable from any
other—they're all just dots (not even that, really).
A vector, on the other hand, includes its own context:
it always specifies a direction and an amount. That's what it is. A vector
that points 80 units across and 60 units down is notably different from
one pointing 20 units across and 30 down. They are different vectors.
So, we can say that each array of numbers defines
a vector. An array of numbers is a vector.
Adding Vectors
It may be obvious that you add two vectors simply by
adding their component values:
[20,70] + [60, -10] = [80, 60]
This addition can be represented graphically, by placing
the vectors end-to-end:
It follows that every vector can be broken down into
a set of vectors representing its component values:
[80, 60] = [80,0] + [0, 60]
When you represent this addition graphically, you see
that the components of 2D vectors make right triangles:
We'll use this later to apply to vectors certain properties
of right triangles, such as the Pythagorean Theorem (which we discuss
in the next section).
Subtracting Vectors
You subtract two vectors by subtracting their respective
components:
[80, 60] - [20, 70] = [60, -10]
Subtracting a vector is the same as adding its inverse.
That is:
-[c, d]= [-c,-d]
This is the same as following the vector in the reverse
direction, e.g. instead of down and right, you travel up and left. You
can represent this graphically simply by placing the arrowhead at the
opposite end:
As you can see, we can find the vector between two layers
simply by subtracting one's position from the other's. The distance between
the two layers will be the length of this vector.
Question: How do you subtract one vector from another, for example [80,60] - [20,70]? Answer: You subtract two vectors by subtracting their respective components: [80,60] - [20,70] = [60,-10]. | 677.169 | 1 |
Alternative: Balls can be used instead of blocks, instead drawing a circular "target" on the camera screen, but linear width is an easier way to connect to angular width in the next activity, where strings actually demonstrate the angle which is the size of the object in the image.
Describing the pattern of data in words is especially important, as it sets the stage for the algebraic relationship developed in the next activity.
Use the students' words for "size in the image", holding back on introducing "angular size" until the next activity.
If desired, instructor can take JPEG images of objects (or have students do so), and convert JPEG to FITS, using this online image converter utility: Image Converter.
Question: Why is describing the pattern of data in words important? Answer: It sets the stage for the algebraic relationship developed in the next activity | 677.169 | 1 |
Seven Sided Polygon
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Description
A polygon is a geometrical shape which is constructed by the straight lines. In geometry there
are various kinds of polygons are defined that are constructed with help of straight lines.
Polygon is a special kind of geometrical shape that has equal no.
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seven sided polygon seven sided polygon a polygon is a geometrical shape which is constructed by the straight lines in geometry there are various kinds of polygons are defined that are constructed with help of straight lines polygon is a special kind of geometrical shape that has equal no of line segment and same amount of connecting points in mathematical definition we can describe the polygon as a geometrical shape which is enclosed with the several line segments in the polygons the connecting points are popularly called as vertex point and in the same aspect the line which connects the two vertex point to each other are known as edges of the polygon in the polygon there is no limit for no of line segment that are exists into the polygon in the formation of polygon it is necessary that to form the polygon with three sides.according to above definition of polygon we can say that triangle is a geometrical shape which can be consider as a polygon because triangle is a shape which is formed by three straight lines normally in the concept of polygon we discuss about the regular polygon regular polygon is a type of polygon in which all the sides and all the angles are equal in the measure know more about how to divide decimals math.tutorvista.com page no 1/4
p. 2
Question: What are the points where lines of a polygon meet called? Answer: Vertex points | 677.169 | 1 |
Spherical Coordinates
FlexibleBandana8967 asked
The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates ?, ? , as follows. We take the origin to be the center of the Earth and the positive z-axis to pass through the North Pole. The positive x-axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of P is ? = 90° - and the longitude is ? = 360° - ?°. Find the great-circle distance from town A (lat. 45.52° N, long. 85.14° W) to town B (lat. 41.25° N, long. 118.25° W). Take the radius of the earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere. Round your answer to the nearest mile.)
Answers (1)
Answered by Anonymous4 minutes later
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For 9th grade science, know that there are other co-ordinate
systems that can be used not just azimuth and altitude, there is
also a celestial coordinate system, cartesian coordiante system but
all of them need two references, for example in a cartesian
coordiante system we need x and y coordiantes, in celestial
coordianate system we need right assention and declination.
So we need to know how far across it is and how far up it is at the
very least. This is the minimum amount of information necessary to
describe the position of an object.
Actually in 3 dimensional space we need 3 measurements, for example
azimuth, altitude and distance from point of origin.
Can you just accept that this is true or at least think hard about
how you would find a co-ordinate system that had only one
coordiante and how you would describe the position of an object in
2 or 3 dimensions?
A single measurement in more advanced level physics and mathematics
is known as a scalar value, it gives you a number and nothing more.
Let's say you want to know where I live to come and visit, but I
can only tell you where I live in scalar units. I tell you I live
exactly 110 miles from your house. That's all the information I can
give you. How do you go about finding me? You see how useless that
information is? Now if I told you I lived exactly 110 miles due
north of where you live, now you have a direction and a distance
and you can find me - see why you need two co-ordiantes?
If you can't accept this wait until you get to college level linear
algebra then you can have a formal mathematical proof of why you
need at least n coordinates to describe a point in n dimensions.
Question: Which of the following is a type of coordinate system mentioned in the text? A) Cartesian B) Celestial C) Polar D) All of the above Answer: D) All of the above
Question: Why is a single measurement (scalar value) insufficient to describe the position of an object in 2D or 3D space? Answer: A single measurement only provides a number and not the direction, making it impossible to determine the exact location of an object. | 677.169 | 1 |
Trapezoid
It is necessary that the two parallel sides be opposite; they cannot logically be adjacent.
If the other pair of opposite sides is also parallel, then the trapezoid is a parallelogram.
(But according to some authorities, parallelograms are specifically excluded from the definition of trapezoid.)
Otherwise, the other two opposite sides may be extended until they meet at a point, forming a triangle that the trapezoid lies inside of.
The area of a trapezoid can be computed as the product of the distance of the two parallel sides and the average (arithmetic mean) of the other two sides. This yields the well-known formula for the area of a triangle, were one to consider a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.
In acrobatics, the trapeze is a certain acrobatic device that is shaped like a trapezoid
Question: What happens when the other two opposite sides of a trapezoid are extended? Answer: The other two opposite sides may be extended until they meet at a point, forming a triangle that the trapezoid lies inside of. | 677.169 | 1 |
Hello,
I'm building a google maps application, and I need to do some
calculation with polygons. I have several points in cartesian system
(lat/lng)
that I use to create the polygons. I need to remove some of this points,
because they are irrelevants (i.e. are too close to each other, or is a
point
in the middle of a line, so it could be removed for sure) and are consuming
memory vainly.
So, the logic that I thinked was to get three of the points of the polygon,
and taking the middle point as the origin point, and the two others to
make the lines, and then calc the angle between this two lines. if it is
close to 180 degrees,
i can remove the middle point. And so I would do this sequentially.
I couldn't find a postgis function that can return me the angle between two
lines,
so any suggestions???
Thank you all
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Question: What is the user's plan after removing the irrelevant points? Answer: To save memory by reducing the number of points in the polygons. | 677.169 | 1 |
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