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Triangle Trilemma (Java)
Today's exercise is Problem A from the Google Code Jam Beta 2008. The problem is to accept three points as input, determine if they form a triangle, and, if they do, classify it at equilateral (all three sides the same), isoceles (two sides the same, the other different), or scalene (all three sides different), and also classify it as acute (all three angles less than 90 degrees), obtuse (one angle greater than 90 degrees) or right (one angle equal 90 degrees).
Your task is to write the triangle classifier.
Solution:
Plot three points on the (x,y) plane and join them with lines. If the points are not in the same straight line, you will have created a triangle.
Internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':
Distance can be calculated using:
After calculating three distances, internal angels 'A','B' and 'C' can be calculated using these formula:
Step 1: Begin by using the cosine rule to find the largest angle.
Note: The largest angle is always opposite to the largest side (assumed to be distance 2 here):
Step 2: Use the sine rule to find one of the remaining angles.
Step 3: Use the 'sum of internal angles' rule to find the third angle.
Question: What is the third step in calculating the internal angles of a triangle? Answer: The third step is to use the'sum of internal angles' rule to find the third angle. | 677.169 | 1 |
And statement 2) says that the area of our slice/sector is 4.5π, or 4.5/36 = 1/8 the total area of the whole pie. Once again, an eighth is an eighth is an eighth, so by the exact same reasoning we get 45 degrees. Sufficient!
This is a pretty basic rule, but it's widely applicable to many circle problems. It can be especially useful with central triangles, which have central angles by definition. Take a look at today's question of the day to test your skills, and when you get it right, treat yourself to a slice of pepperoni as a reward!
Question:
What is the area of the circle above with center O?
(1) The area of D AOC is 18.
(2) The length of arc ABC is 3π.
Answer:
Step 1: Analyze the Question Stem
This is a Value question. For any Circle question, we
only need one defining parameter of a circle (area,
circumference, diameter, or radius) in order to calculate
any of the other parameters. Also, all radii of the same
circle will have the same length. So AO = CO. That makes
the triangle an isosceles right triangle (or 45-45-90) for
which we know the ratio of the sides. Not only would the
circle's circumference, diameter, or radius be sufficient,
but information that gave us any side length of triangle
AOC would also be sufficient, as it would give us the length
of the radius.
Step 2: Evaluate the Statements
Statement (1): We are given the area, and we already know
that the base and the height are equal. So if we call the
radius of the circle r, then the area of the triangle is equal to
1/2 (Base)(Height) = 1/2 (r)(r) = 1/2 r ^2 = 18
Remember that we are not asked to calculate the actual
value of r. Because we have set up an equation with
one variable, and we know in this case that r can only
be positive since it is part of a geometry figure, we have
enough information to determine the area of the circle (the
solution would have been r ^2 = 36, r = 6).
Therefore, Statement (1) is sufficient. We can eliminate
choices (B), (C), and (E).
Statement (2): Because O is the center of the circle and
angle AOC measures 90 degrees, we know that the length
of arc ABC is one-fourth of the circumference. Because this
statement allows us to solve for the circumference, it is
sufficient.
Question: What is the length of arc ABC if the radius of the circle is 6? Answer: 18π
Question: What is the area of the sector AOC if the radius of the circle is 6? Answer: 27π
Question: What is the area of the whole pie (circle)? Answer: 144π | 677.169 | 1 |
Once you have all of the information written on your figure (that you, of course, have redrawn in your scratch work), start to look for familiar shapes hidden in the multiple figures. These shapes can be triangles, quadrilaterals or any other shape that will all you to solve. If the problem features variables, the shape will usually involve the parts of the figure in which they show up.
On the most advanced GMAT content, you will see complicated geometric figures. While these questions are often the most daunting questions, you (as the prepared and confident test taker) will realize that the GMAT is only testing a couple concepts.
The geometric concepts you need to know on test day include the equations for finding the area, circumference, and volume of triangles, circles, and quadrilaterals. That is it. Nothing else. While those are straight forward topics, Multiple Figures on the GMAT push your understanding of these concepts to the extreme.
When presented with a complex image that doesn't easily fit into the category of circle, triangle, or quadrilateral, you want to proceed by following the check list below to identify the relevant component of the image that will unlock the simplicity mentioned above:
Are there parallel lines?
This may be one of the most helpful questions you can ask. If there are parallel lines, you can deduce a significant amount of information from the properties of parallel lines and the subsequent angles of the intersecting lines.
Are there recognizable angles?
As you are studying for the GMAT, you will come across several common triangles that conform to common, simplistic structure, including 30-60-90 degree triangles and 45- 45- 90 triangles. These angles have important properties that can unlock the right answer.
If it is circle, can I figure out the radius or diameter?
Of course, if you can calculate either the radius or the diameter, you can calculate the other. However, for any image containing a circle, the radius is the key to solving the problem. Identify creative ways of locating it.
Do I know the internal angles?
The internal angles of triangles sum to 180. The internal angles of quadrilaterals sum to 360. Can you backsolve with these totals into the correct answer?
Consider the above four points a simple checklist to approach the most difficult GMAT geometry content. Many times, you'll be required to employ several of the points above to uncover the right answer. These represent just the points you want to start evaluating.
Remember, you have to memorize certain formulas and be able to apply them to complex situations. The combination of memorization and strategy together is the key to a fantastic score. Good luck on geometry!
After the radius, the most important number to understand is π. π is defined as the ratio between the circumference of a circle and its diameter. Thus, the formula for finding the circumference of a circle is 2πr. You should also know the formula for the area of a circle, which is πr2.
Question: What is the ratio that defines π? Answer: The ratio between the circumference of a circle and its diameter
Question: What are the main geometric concepts one should know for the GMAT? Answer: The equations for finding the area, circumference, and volume of triangles, circles, and quadrilaterals.
Question: Which of the following shapes could be used to solve a GMAT problem? A) Pentagon, B) Ellipse, C) Triangle, D) Both A and B. Answer: C) Triangle
Question: What is the first step in approaching complex geometric figures on the GMAT? Answer: Identify if there are parallel lines in the figure. | 677.169 | 1 |
protractor
protractor, any of a group of instruments used to construct and measure plane angles. The simplest protractor comprises a semicircular disk graduated in degrees—from 0° to 180°. It is an ancient device that was already in use during the 13th century. At that time, European instrument makers constructed an astronomical observing device called the torquetum that was equipped with a semicircular protractor.
A more complex form of protractor, designed for plotting the position of a ship on navigational charts, was invented in 1801 by Joseph Huddart, a U.S. naval captain. This instrument, called a three-arm protractor, or station pointer, is composed of a circular scale connected to three arms. The centre arm is fixed, while the outer two are rotatable, capable of being set at any angle relative to the centre one.
A related instrument used by marine navigators is the course protractor. It provides an effective tool with which to measure the angular distance between magnetic north and the course plotted on a navigational
Question: What is the range of degrees on a simple protractor? Answer: 0° to 180° | 677.169 | 1 |
Thursday 10/11/12
Lesson 2.5 Day 1 worksheet. Remember to go back to your text book and review the examples in 2.5 that use the segment addition postulate and the angle addition postulate.
Friday 10/12/12
Study the example(s) completed in class, on the Lesson 2.5 Day 2 work sheet. On the back of the sheet, do the proof using different logic, using the hints provided. Carefully read the Chapter 2 project sheets, to decide which project you want to do. Test on Lessons 2.2 - 2.5 on Tuesday 10/16. The retest of the take home portion of the unit 1 test is due on Monday.
Monday 10/15/12
Prepare for Test on Lesson 2.2 - 2.5 tomorrow. Practice! Finish or redo any worksheets from class. Use the on-line text book resources!
Try the proofs on the Lesson 2.6 Day 2 packet - remember to solve the problem in your head first, and developing an overall stategy before you write anything down! If you find some of the proofs frustrating or difficult, don't panic - we'll take care of it in class. However, if you are struggling, do some problems from the text book and check your answers.
Finish any proofs your were not able to complete in class. Test on 2.6 and 2.7 on Friday. RTN & GP Lesson 3.1. All of the highlighted vocabulary terms on page 141 should be familiar to you. Make sure that you realize that these are angle relationship names based on position only, and do not tell you anything about congruence or measures! The two new postulates should make intuitive sense to you.
Thursday 10/25
Prepare for test on 2.6 and 2.7.
Friday10/26/12
Page 142: #3, 4, 5, 6, 15, 16, 17, 28, 29, 30, 31, 32
RTN & GP Lesson 3.2
Monday 10/29/12
Page 150: #22 – 36. For each problem (except #34), copy the diagram, set up an equation and state the postulate or theorem that justifies it, using the acceptable abbreviation.
RTN & GP Lesson 3.3
Monday 11/12/12
WELCOME BACK FROM HURRICATION 2012! No HW
Tuesday 11/13/12
Do pages 1 & 2 ONLY of the 3.2/3.3 packet
RTN & GP Lesson 3.3
Wednesday 11/14/12
Question: What should students do if they find some proofs frustrating or difficult on Monday 10/15/12? Answer: If they find some of the proofs frustrating or difficult, they should not panic and should try some problems from the textbook to check their answers. | 677.169 | 1 |
Geometry
posted on: 09 Dec, 2011 | updated on: 15 Jan, 2013
Geometry is defined as the branch of mathematics which is concerned with the studies of the problems related to shape, size, position of figures, construction, measurement of angles sides, etc. In other words, it can be said as a science concerned with the studying of lengths, area and volumes.
There are various topics covered under geometry. The very basis of geometry is a Point. It may be defined as a dot or the exact location. The next part of geometry covers the angles and the lines. There are various Types of Angles called as the Right Angle, obtuse angle, acute angle, reflex angle, straight angle to name a few. Then we move on to the various kinds of lines, the basic lines are the Parallel Lines, perpendicular lines, etc.
Geometry is considered as the basic study concerned with surveying, measurements, area and volume. Geometry is a very vast study of mathematics. It shows a relationship between shapes and mathematics. The very important part of geometry is also construction, which is related to construction of angles, triangles, quadrilaterals, circles and many more.
Geometry is also a study that deals with the deduction of the properties, measurement, lines angles etc. Geometry generally began with the measurement of things. Geometry basically means to "measure the earth" and is the science of shapes and size of things.
Geometry is a very important part of our everyday life, like if we need to carpet our floor an do white washing as well for doing this we need to take out the area of the floor first, thus to proceed with this we need to apply geometry i.e. find the area, so we can easily use it in our day to day life. We now have detailed knowledge about "Geometry".
Question: What is the relationship between geometry and construction? Answer: Geometry is used in construction to construct shapes like angles, triangles, quadrilaterals, and circles. | 677.169 | 1 |
Angle PCQ = 360 - 60 - 90 - 60 = 150. Sides are CQ and CP
So we have 3 congruent triangles. Hence the thrid side of each is the same length. And the triangle composed of those three sides must be equilateral.
Not so for parallelogram
You can put this solution on YOUR website! You did. well done.
Use this to check if you need to
Graphs/147472: For each of the following inequalities indicate whether the boundary line should be dashed or solid.
x+y<2 2x<-y
2x-y>3 x+y>1
y>2 4x-2y+1>0
2>x-y
Quadratic_Equations/147469: Let's put an equation with that curve. Use y = x2-8x+7
a. What is the value of the discriminant? Review Lesson 3.05.
b. What types of roots will this equation have? Ex: two complex roots, one repeated root, two real rational roots or two real irrational roots? Review Lesson 3.05.
c. Find the roots by factoring and solving. Review Lesson 3.04.
d. Find the roots by using the quadratic equation. Review Lesson 3.04.
e. Find the roots by graphing and using the tracking tool. 1 solutions Answer 107856 by vleith(2825) on 2008-07-07 18:50:49 (Show Source):
You can put this solution on YOUR website! I think that's a trick question. Sure it will ... if you park it at a 60 degree angle.
If you have to park it 'straight', then we have to do a little math.
Draw your parallelogram, 45 feet and 30 feet with 60 degree angle between the 30 and 45 sides.
Drop a line from one end of the a 45 foot side down. Make it perpendicular to the 30 foot base line.
Now take 45 *cos(60) to see how long the side of the resulting right triangle is. Take 45*sin(60) to see how long the other side is. Then make the call about whether the rv can fit
expressions/147466: a mechanic earns a salary of $350 a week plus he gets $10 for every oil change he carries out.
a) write an eqation to represent the total weekly earnings E, when C oil changes are done.
Question: How many congruent triangles are mentioned in the text? Answer: 3 | 677.169 | 1 |
Last week, we started our triginometry unit in geometry. We have been using the Sin, Cos, and Tan formulas in class and on the homework for the last few classes. The trig formulas are really interesting, because hey allow you to find a side length of a right triangle, when you only know the measures of a side and two angles. Its also cool that for eah angle measure, the ratio is always the same, no matter how big or small the triangle is. This makes finding the ratios so much easier!
Last week in geometry, we talked about, and did a lot of work with circles. We learned the formulas to find the area and circunfrance of a circle. We also learned a lot about the radius and diameter of a circle , and their special properties, and their relationships with pi. As well as learning about circles, we learned how to find the area of regular polygons, trapezoids, kites, and rhombuses. We also had a test at the end of the week. I thought it was going to be a lot harder than it actually was.
There are a few tricks you can use in certain right triangles to find out the lengths of all the sides if you only know the length of one side. The triangle either has to have two 45 degree angles, and a right angle; or it can have 30, 60, and 90 degree angles. In the 45, 45, 90 triangle; the hypotenuse is the length of one of the legs times the square root of two. in the 30, 60, 90; the long leg is the short leg times the square root of three, and the hypotenuse is twice the length of the short leg.
The pythagorean theorem is used to find a leg, or the hyptenuse of a right triangle, when you have the measurements of two sides. The theorem is cool, because you can use it for a lot of different purposes. Since a lot of different shapes are made up of triangles, like parallelagrams, you can use the pythagorean theorem to help you find the area or perimiter of those shapes.
Last week in geometry we learned about radicals. Sometimes the square roots of numbers are never ending, unrepeting decimals; so we use simplified radicals to write them instead. Radicals can sometimes take a while to simplify, but i think that once i memorize a lot of the perfect squares, it will be much easier.
One week in geometry we learned about reflections. Reflections are when a shape is reflected across a line. This makes the line of reflection very similar to a line of symmetry. Reflections can sometimes be difficult, because ou have to reflect each point seperately, and if you mess one up, the whole thingis wrong; although it usually easy to notice.
That week in geometry we learned about translations, dialations and rotatins, which are all forms of transformation. Translations are when the hape is moved, dialations are when the shape changes size, and are when the shape is rotated around one point.
Question: What is the special property of the ratio in trigonometry for a right triangle? Answer: The ratio is always the same, no matter how big or small the triangle is
Question: What are some of the shapes that the Pythagorean theorem can be used to find the area or perimeter of? Answer: Parallelograms
Question: What is the name of the theorem used to find a leg or the hypotenuse of a right triangle when you have the measurements of two sides? Answer: The Pythagorean theorem
Question: What are the two types of special right triangles mentioned in the text? Answer: 45-45-90 triangles and 30-60-90 triangles | 677.169 | 1 |
Figure Solution To Isometric Drawing Problem 1
b. Slanted and Oblique Surfaces. Figure 40 (on the following page) is a sample problem that involves the creation of an isometric drawing from given orthographic views that contain a slanted surface. The slanted surface is dimensioned by using an angular dimension. That presents a problem because angular dimensions cannot be directly transferred from orthographic views to isometric drawings.
Question: What is the main problem presented by the use of angular dimensions in this context? Answer: That it presents a problem because angular dimensions cannot be directly transferred from orthographic views to isometric drawings. | 677.169 | 1 |
The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greekἀγκύλος(ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[2]
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.[3]
Measuring angles
The size of an angle is characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are called congruent angles.
In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.
The measure of angle θ is the quotient of s and r.
In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the arc r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):
The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.
Units
Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.
Question: What does the Greek word 'ἀγκύλος' mean? Answer: The Greek word 'ἀγκύλος' means "crooked, curved".
Question: Which units are most commonly used to represent angles? Answer: The degree and the radian are by far the most commonly used units to represent angles.
Question: Which mathematician regarded an angle as a deviation from a straight line? Answer: Eudemus | 677.169 | 1 |
The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree = 1/21600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.
Positive and negative angles
Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.
In a two dimensional Cartesian coordinate system, angles are typically defined relative to the positive x-axis with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented as they commonly are, with the x-axis rightward and the y-axis upward, positive rotations are counterclockwise and negative rotations are clockwise.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as − 45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. However, a rotation of − 45° would not be the same as a rotation of 315°.
In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearings are measured relative to north. By convention, viewed from above, bearing angle are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
Alternative ways of measuring the size of an angle
There are several alternatives to measuring the size of an angle by the corresponding angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.
Question: In a 2D Cartesian coordinate system, which direction do positive angles represent? Answer: Counterclockwise
Question: In navigation, what is a bearing of 45° equivalent to? Answer: North-east orientation | 677.169 | 1 |
In rational geometry the spread between two lines is defined at the square of sine of the angle between the lines. Since the sine of an angle and the sine of its supplementary angle are the same any angle of rotation that maps one of the lines into the other leads to the same value of the spread between the lines.
Astronomical approximations
Astronomers measure angular separation of objects in degrees from their point of observation.
1° is approximately the width of a little finger at arm's length.
10° is approximately the width of a closed fist at arm's length.
20° is approximately the width of a handspan at arm's length.
These measurements clearly depend on the individual subject, and the above should be treated as rough approximations only.
Identifying angles
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, ...) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is typically not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples.
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").
Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.
Angles equal to 1/2 turn (180° or two right angles) are called straight angles.
Angles equal to 1 turn (360° or four right angles) are called full angles.
Angles that are not right angles or a multiple of a right angle are called oblique angles.
Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
Question: What is the definition of the spread between two lines in rational geometry? Answer: The spread between two lines is defined as the square of the sine of the angle between the lines.
Question: Which Greek letter is typically not used to represent an angle in mathematical expressions? Answer: π | 677.169 | 1 |
The angle between the two curves at P is defined as the angle between the tangents A and B at P
The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.[8]
In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.
Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio
Question: True or False: The small-angle formula can be used to convert an angular measurement into an actual distance. Answer: False. It can be used to convert into a distance/size ratio. | 677.169 | 1 |
English What is the syntax in,bending, I bow my head and lay my hand upon her hair, combing and think how do woman do this for each other.
Math The endpoints of one diagonal of a The The endpoints of one diagonal of a rhombus are (0, -8) and (8, -4). If the coordinates of the 3rd vertex are (1, 0), what are the coordinates of the 4th vertex? (7, -12) (7, -8) (-8, -4) (-4, -12)
science-VERY URGENT Hummingbirds. They have a huge beak thing to poke down the long tube. And they love red. Mice come out at night Bees have to land
Biology Probably rough endoplasmic reticulum. An enzyme is a protein. You call it rough because it is studded with ribosome units. Ribosomes make proteins. More protein "factories" equals more enzyme to be released Note: Lysosome's are were you send proteins to be destroyed
Question: What are the coordinates of the 4th vertex of the rhombus? Answer: (-4, -12) | 677.169 | 1 |
Meanwhile, area QCR is a triangle whose base is the line segmentQC of length ae, and whose height is asinE:
Combining all of the above:
Dividing through by a2 / 2:
To understand the significance of this formula, consider an analogous formula giving an angle θ during circular motion with constant angular velocity M:
Setting M = 2π / τ and
θ = E - esinE gives us Kepler's equation. Kepler referred to M as the mean motion, and E - esinE as the mean anomaly. The term "mean" in this case refers to the fact that we have "averaged" the satellite's non-constant angular velocity over an entire period to make the satellite's motion amenable to analysis. All satellites traverse an angle of 2π per orbital period τ, so the mean angular velocity is always 2π / τ.
Substituting M into the formula we derived above gives this:
This formula is commonly referred to as Kepler's equation.
Application
With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of θ from periapsis is broken into two steps:
Compute the eccentric anomaly E from true anomaly θ
Compute the time-of-flight T from the eccentric anomaly E
Finding the angle at a given time is harder. Kepler's equation is transcendental in E, meaning it cannot be solved for E analytically, and so numerical approaches must be used. In effect, one must guess a value of E and solve for time-of-flight; then adjust E as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence.
The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity e is nearly 1, and plugging e = 1 into the formula for mean anomaly, E - sinE, we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it doesn't hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.
Perturbation theory
You can deal with perturbations just by summing the forces and integrating, but that's not always best. Historically, people (who?) did variation of parameters, which works better in some ways.
Modern techniques
Today, we don't use the same techniques that Kepler used, in general.
Conic orbits
For simple things like computing the delta-v for coplanar transfer ellipses, traditional approaches work pretty well. But time-of-flight is harder, especially for nearly-circular orbits, or for hyperbolic orbits.
Transfer Orbits
Question: What is the universal variable formulation? Answer: The universal variable formulation is a method developed to address the convergence issues and limitations of Kepler's equation for extreme elliptical orbits and non-elliptical orbits (parabolic or hyperbolic). | 677.169 | 1 |
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Sunday, 21 March 2010
Line Symmetry: video, quizzes, and online jigsaw puzzle
Last updated: 22 March 2010
Symmetry is covered in different subjects, for example, mathematics, science, technology, and humanities, but for the majority of us, the most familar form of understanding symmetry is geometrical symmetry. Within geometrical symmetry, there are also different types, the most familiar being line symmetry and rotational symmetry.
This post will deal only with 2-dimensional line symmetry.
If we draw an imaginary line across an object, and one side of it is the mirror image of the other, the object is said to have line symmetry; this imaginary line is called the 'line of symmetry'. Line symmetry is also known as reflection symmetry, mirror symmetry, mirror-image symmetry and bilateral symmetry.
Some objects, such as a butterfly, have only one line of symmetry; others have more than one.
We can see many examples of symmetry around us every day, for example, animals, traffic signs, cars, buildings and, of course, the human form itself.
Watch this video on symmetry by clicking on the image below. Due to copyrights issues, I am unable to embed the video here, but when you click on the image you will be directed to the video on Youtube.
Put your knowledge of symmetry to the test with this activity on geometric shapes. Click here to begin.
And now, try this quiz. Your email is required; you will get the results together with the corrections sent to you. Click here to begin.
And if you enjoy a jigsaw puzzle challenge, try this 60-piece Taj Mahal puzzle. Click here to begin.
When you've done all the activities, please come back here to comment. Thanks. Have fun learning!
Question: Which of the following is NOT a term for line symmetry? A) Reflection symmetry B) Translational symmetry C) Mirror symmetry Answer: B) Translational symmetry
Question: What are some everyday examples of line symmetry? Answer: Animals, traffic signs, cars, buildings, and the human form. | 677.169 | 1 |
You can put this solution on YOUR website! With not just one but both equations already "solved for y", this system is an ideal candidate for using the Substitution Method. The first equation says the y = 2x. Substituting 2x for the y in the other equation we get:
2x = -x + 3
Now we solve for x. Adding x to each side:
3x = 3
Dividing by 3 we get:
x = 1
Now we use this value for x and either of the original equations to find the y value:
y = 2(1)
So y = 2
The solution is the point (1, 2)
Triangles/717073: if triangle abc is congruent to triangle tuv which statement is incorrect?
angle c is congruent to angle v
line bc is congruent to line uv
abc is congruent to utv
line ab is congruent to line tu 1 solutions Answer 440084 by jsmallt9(3296) on 2013-02-21 08:42:32 (Show Source):
You can put this solution on YOUR website! When you're told that triangle ABC is congruent to triangle TUV, you are told more than just the fact that the triangles are congruent.
The letters of the names of triangles represent vertices of the triangle. And usually the order of the letters is unimportant. But when a statement says that two triangles are congruent the order of the letters means something important. The first letters in the names of the triangle are corresponding vertices; the second letters are corresponding vertices and the third letters are corresponding vertices. Since corresponding parts of congruent triangles are congruent we know that if triangle abc is congruent to triangle tuv then:
Angle A is congruent to angle T
Angle B is congruent to angle U
Angle C is congruent to angle V
Side AB (which is between the first two vertices) is congruent to side TU
Side BC (which is between the second and third vertices) is congruent to side UV
Side AC (which is between the first and third vertices) is congruent to side TV
Except for "ABC is congruent to UTV" you should now what the answers are. For "ABC is congruent to UTV" (assuming this means "triangle ABC is congruent to triangle UTV") then this is false because the corresponding parts don't match up. If it had been "triangle BCA is congruent to triangle UVT", then it would be true because the same letters correspond with each other: B and U, C and V and A and T
P.S. Points should be named with capital/upper-case letters. And vertices of a triangle are just special points so they also should be named with capital letters.
You can put this solution on YOUR website! What does "log under root" mean?? You'll have to learn how to read your problems correctly so you can then describe them correctly when you post.
Question: Which method is used to solve this system of equations? Answer: The Substitution Method.
Question: What is the relationship between the sides AB and TU in congruent triangles ABC and TUV? Answer: Side AB is congruent to side TU.
Question: If triangle ABC is congruent to triangle TUV, which of the following statements is incorrect? A) Angle A is congruent to angle T. B) Angle B is congruent to angle U. C) Angle C is congruent to angle V. D) Side AC is congruent to side TV. Answer: D) Side AC is congruent to side TV (The incorrect statement is that sides AC and TV are congruent, but it's the vertices that are corresponding, not the sides between them). | 677.169 | 1 |
constructibility of the n-gon for any n that is a prime of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
also known as Fermat primes.
One of the nicest actual constructions of the 17-gon is
Richmond's (1893), as reproduced in Stewart's Galois Theory.
Draw a circle centred at O, and choose one vertex V on the circle. Then
locate the point A on the circle such that OA is perpendicular to OV,
and locate point B on OA such that OB is a quarter of OA. Next locate
the point C on OV such that angle OBC OBC Other Backward Classes OBC Ontario Building Code OBC On Board Computer OBC Organization for Bat Conservation OBC Outline Business Case (UK government procurement) OBC Oriental Bank of Commerce (India) is a quarter the angle OBV OBV Obviously OBV On Balance Volume (market momentum indicator that relates volume to price change; developed by Joseph Granville) OBV Orbital Boost Vehicle OBV On Board Video OBV Obligated Volunteer . Then
find the point D on OV (extended) such that angle DBC See dBA. (language, parallel) DBC - A data-parallel bit-serial C based on MPL. SRC, Bowie MD.
E-mail: <[email protected]>. is half of a right
angle. Let E denote the point where the circle on DV as diameter cuts
OA. Now draw a circle centred at C through the point E, and let F and G
denote the two points where this circle cuts OV. Then, if perpendiculars
to OV are drawn at F and G, they strike the main circle (the one centred
at O through V) at points [V.sub.3] and [V.sub.5], as shown in the
figure.
[ILLUSTRATION OMITTED]
The points V, [V.sub.3], and [V.sub.5] are the zero, third, and
fifth vertices The plural of vertex. See vertex. of a regular heptadecagon, from which the remaining
vertices are easily found (i.e., bisectbi·sect v.bi·sect·ed, bi·sect·ing, bi·sects
v.tr. To cut or divide into two parts, especially two equal parts.
v.intr. To split; fork. angle [V.sub.3]O[V.sub.5] to
locate [V.sub.4], etc.). Gauss was clearly fond of this discovery, and
there is a story that he asked to have a heptadecagon carved on his
t , like the sphere in a cylinder on Archimedes'
tombstone.
Useful links and references
Eves, H. (1990). An Introduction to the History of Mathematics,
(6th ed.). Saunders Publishing Co.
Stewart, I. (2004). Galois Theory Chapman
& Hall.
Question: What method is used to locate the remaining vertices of the heptadecagon after finding V, V3, and V5? Answer: Bisecting the angle V3OV5 to locate V4, etc. | 677.169 | 1 |
I have a ladder that is 55' in length leaning against a wall, which I believe is the hypotenuse. The angle where the ladder meets the street is 55 degrees. I need to find the length of the other two sides of the triangle. I wasn't good at math 30 years ago and it hasn't gotten any better. Thanks in advance for your help.
Best regards,
ML Click here to see answer by bayners123(12)
Question 414735: Choose the statement that is true about the given quantities.
Column A Column B
Sine 45 degrees Cosine 45 degrees
1.) The quantity in column A is greater.
2.) The quantity in column B is greater.
3.) The two quantities are equal.
4.) The realtionship cannot be determined from the given info.
Please help, I don't know what to do! Click here to see answer by sudhanshu_kmr(1152)
Question: What is the length of the ladder? Answer: 55 feet | 677.169 | 1 |
Parts of a Cone
Date: 04/18/2001 at 12:59:14
From: Brian McCormick
Subject: Parts of a solid cone
Hello,
I am a second grade teacher and we are currently teaching a unit on
shapes. The question came up as to whether or not a solid cone has any
edges. My contention is that the definition of an edge is where two
planes intersect, and therefore a cone cannot have an edge. Another
teacher says that the curved surface of a cone represents an infinite
number of planes, and therefore represents an infinite number of
edges.
I would very much appreciate your response, and don't be afraid to get
technical. This is as much to satisfy my own curiosity as to let the
kids know the proper answer.
Brian McCormick
Date: 04/18/2001 at 14:25:21
From: Doctor Peterson
Subject: Re: Parts of a solid cone
Hi, Brian.
We get this question from time to time, and can never really give a
definite answer. The word "edge" is used in different ways; often
people get in trouble by introducing the concept of "edge" in the
context of polyhedra (where it does mean the intersection of two flat
faces), but then talking about curved surfaces like cones without
additional comment.
Here's the definition in the Academic Press Dictionary of Science
and Technology:
1. in graph theory, a member of one of two (usually finite) sets
of elements that determine a graph; i.e., an element of the edge
set. The other set is called the vertex set; each element of the
edge set is determined by a pair of elements of the vertex set...
2. a straight line that is the intersection of two faces of a
solid figure.
3. a boundary of a plane geometric figure.
In the latter sense (which I think is appropriate in discussing a
cone, even though the dictionary only mentioned plane figures and not
curved surfaces), the cone has one edge. I definitely would not bring
in the idea of "an infinite number of edges"; that kind of reasoning
generally leads to trouble! I would simply say that we can extend the
concept of edge either from the world of polyhedra (definition 2) or
from the world of plane geometry (definition 3) to apply to possibly
curved boundaries of possibly curved surfaces, as long as we say that
we are doing so. This also agrees with definition 1, which likewise
does not require straightness (indeed, there is no such concept in
graph theory), and which relates to boundaries when we consider planar
graphs (as in Euler's polyhedral formula).
What definition you use depends on what you are going to do with it.
If you are just describing objects, my loose definition is fine. If
you are going to prove theorems involving planes and angles, you'll
want to restrict yourself to the polygonal definition, but then you
Question: How many edges does a cone have according to Doctor Peterson? Answer: A cone has one edge.
Question: What does Doctor Peterson suggest when choosing a definition of an edge? Answer: Doctor Peterson suggests choosing a definition based on what you plan to do with it - whether you're just describing objects or proving theorems involving planes and angles.
Question: Which definition from the Academic Press Dictionary of Science and Technology does Doctor Peterson use to describe an edge in a cone? Answer: Doctor Peterson uses definition 3: "a boundary of a plane geometric figure." | 677.169 | 1 |
A parallelogram is a quadrilateral whose opposite sides
are parallel. The figure below shows an example:
Parallelograms have three very important properties:
Opposite sides are equal.
Opposite
angles are congruent.
Adjacent
angles are supplementary (they add up to 180º).
To visualize this last property, simply picture the opposite
sides of the parallelogram as parallel lines and one of the other
sides as a transversal. You should then be able to see why this
would be true.
The area of a parallelogram is given by the formula:
where b is the length of the base and h is
the height.
The next three quadrilaterals we'll review—rectangles,
rhombuses, and squares—are all special types of parallelograms.
Rectangles
A rectangle is a quadrilateral whose opposite sides are
parallel and whose interior angles are all right angles. A rectangle
is essentially a parallelogram whose angles are all right angles.
As with parallelograms, the opposite sides of a rectangle are equal.
The formula for the area of a rectangle is:
where b is the length of the base and h is
the height.
Rhombuses
A rhombus is a quadrilateral whose opposite sides are
parallel and whose sides are of equal length.
The area of a rhombus is:
where b is the length of the base and h is
the height.
Squares
A square is a quadrilateral in which all the sides are
equal and all the angles are right angles. It is a special type
of rhombus, rectangle, and parallelogram.
The area of a square is:
where s is the length of a side of the
square. The perimeter of a square is 4s.
Solving Polygons by Using Triangles
Polygons can often be cut into triangles, and if you can
solve those triangles, then you can solve the entire polygon. For
example, if you split a square on its diagonal, it forms two 45-45-90
triangles.
Since the hypotenuse of a 45-45-90 triangle always exists
in the ratio of : 1 in relation
to its sides, if you know the sides of a square, you therefore also
always know the measure of its diagonal (d = s). This is simply one example of
the value gained by thinking about a polygon in terms of the triangles
that form it. There are many other ways to use triangles when thinking
about polygons. You can divide a trapezoid or a parallelogram to
make two triangles and a rectangle, or you can draw a diagonal through
a rectangle to make two triangles. In short, as you deal with polygons,
always remember that you can simplify the polygons by cutting them
into triangles.
Question: Can a square be divided into two 45-45-90 triangles? Answer: Yes
Question: What is the difference between a rectangle and a parallelogram? Answer: A rectangle has all interior angles as right angles, while a parallelogram does not necessarily have this property
Question: What is the formula to calculate the perimeter of a square? Answer: Perimeter = 4 × side length | 677.169 | 1 |
Geometry If two six-sided dice are rolled, the probability that they both show the same number can be expressed as a b where a and b are coprime positive integers. What is the value of a+b ?
Thursday, April 11, 2013 at 7:33pm
Geometry(first one is typo) Let ƒÆ=sin −1 7/25 . Consider the sequence of values defined by a n =sin(nƒÆ) . They satisfy the recurrence relation a n+2 =k 1 a n+1 +k 0 a n ,n¸N for some (fixed) real numbers k 1 ,k 0 . The sum k 1 +k 0 can be written as p q , where p...
Thursday, April 11, 2013 at 6:56Thursday, April 11, 2013 at 6:54pm
Geometry ABC is a triangle with circumcenter O, obtuse angle BAC and AB less than AC. M and N are the midpoints of BC and AO respectively. Let D be the intersection of MN with AC. If 2AD=(AB+AC), what is the measure of angle BAC ?
Thursday, April 11, 2013 at 11:27am
Geometry 2x^2-9x-5=0 Is it possible to combine 2x squared with 9x? Not sure how to tackle solving this one.
Wednesday, April 10, 2013 at 9:26pm
geometry Which description does NOT guarantee that a trapezoid is isosceles? A. congruent bases B. congruent legs C. both pairs of base angles congruent D. congruent diagonals
Tuesday, April 9, 2013 at 9:31Monday, April 8, 2013 at 10:29pm
Analytic Geometry/Calculus We didn't go over the perpendicular form in class, only the parallel. Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line.
Monday, April 8, 2013 at 7:40pm
geometry On my last question, just posted, I asked what percent of the area of the target was shaded. The answer options were 5%, 10%, 20% or 80%?
Monday, April 8, 2013 at 6:45pm
geometry There is a square with a star inside. The square has 25cm to show the length of one side. On the right, outside the square, it says 125cm squared, with a line pointing the center of the star in the square. The question is; The target of a shooting game features a shaded star ...
Monday, April 8, 2013 at 6:42pm
Question: Can 2x^2 and 9x be combined in the quadratic equation 2x^2 - 9x - 5 = 0?
Answer: No, they cannot be combined into a single term | 677.169 | 1 |
GEOMETRY The squares of a 3×3 grid of unit squares are coloured randomly and independently so that each square gets one of 5 colours. Three points are then chosen uniformly at random from inside the grid. The probability that these points all have the same colours can be ...
Wednesday, March 27, 2013 at 3:05
Wednesday, March 27, 2013 at 11:58am
Geometry ABCD is a quadrilateral inscribed in a circle with AB = 1, BC = 3, CD = 4 and DA = 6. What is the value of \sec^2 \angle BAD?
Tuesday, March 26, 2013 at 11:34pm
Geometry Determine the number of subsets of A=\{1, 2, \ldots, 10\} whose sum of elements are greater than or equal to 28 .
Tuesday, March 26, 2013 at 11:33pmTuesday, March 26, 2013 at 11:31pmTuesday, March 26, 2013 at 11:31 When writing a math expression, any time there is an open bracket "(", it is eventually followed by a closed bracket ")". When we have a complicated expression, there may be several brackets nested amongst each other, such as in the expression (x+1)*((x-2...
Tuesday, March 26, 2013 at 11:27pm
Geometry When you use the distance formula, you are building a _____ triangle whose hypotenuse goes between two given points. A.right B.equilateral C.acute D.None of these
Tuesday, March 26, 2013 at 10:02am
geometry The volume of a small circle is 10 cubic meters. Find the volume of the larger sphere in cubis meters.
Tuesday, March 26, 2013 at 9:58am
geometry can someone please show me the answer for this problem ? a tree is situated on level ground from a point 135 feet from the base of the tree the measure of the angle of elevation from the ground to the top of the tree is 43 degrees which is the height of the tree to the nearest...
Monday, March 25, 2013 at 1:39pm
CST GEOMETRY a tree is situated on level ground from a point 135 feet from the base of the tree the measure of the angle of elevation from the ground to the top of the tree is 43 degrees which is the height of the tree to the nearest foot?
Monday, March 25, 2013 at 1:23pm
Question: How many subsets of the set A={1, 2,..., 10} have a sum of elements greater than or equal to 28? Answer: The number of such subsets is not calculated in the text. | 677.169 | 1 |
Geometry Six standard six-sided die are rolled. Let p be the probability that the dice can be arranged in a row such that for 1\leq k \leq 6 the sum of the first k dice is not a multiple of 3. Then p can be expressed as \frac{a}{b} where a and b are coprime positive integers. What is ...
Tuesday, March 12, 2013 at 10:04pm
Geometry The Z company specializes in caring for zebras. They want to make a 3-dimensional "Z" to put in front of their company headquarters. The "Z" is 15 inches thick and the perimeter of the base is 390 inches. What is the lateral surface area of this "Z"?
Tuesday, March 12, 2013 at 7:34pm
geometry ABCD is a trapezoid with AB parallel to DC. If AB=25, BC=24, CD=50 and AD=7, what is the area of ABCD?
Tuesday, March 12, 2013 at 6:07pm
geometryTuesday, March 12, 2013 at 6:06pm
GEOmetry.. 7 points are placed in a regular hexagon with side length 20. Let m denote the distance between the two closest points. What is the maximum possible value of m?
Tuesday, March 12, 2013 at 12:12pm
Geometry Point M is the midpoint of the line equation ABMC. Which of the following is not true? (1)AM = MC (2) AB < MC (3) AM > BC or (4) BM < MC
Tuesday, March 12, 2013 at 11:05am
geometry Square ABCD and circle T have equal areas and share the same center O.The circle intersects side AB at points E and F.Given that EF=√(1600-400π),what is the radius of T ?
Tuesday, March 12, 2013 at 9:43am
geometry A pyramid has a square bottom, with an area equal to 64 squares meters. the height of the pyramid is 7 inches. if you start at the top of the pyramid and slide all the way down the middle of one of the sides how many feet will you move?
Tuesday, March 12, 2013 at 2:41am
geometry If ABCDE and LMNOP are similar polygons, then the ratio of AB to LM must be equal to the ratio of CD to NO. (Assume and are corresponding sides, as are and .)
Monday, March 11, 2013 at 2:49pm
Question: What is the thickness of the 3-dimensional "Z"? Answer: 15 inches. | 677.169 | 1 |
Trigonometry
You can use the basic trigonometric identities, along with the definitions of the trigonometric functions, to verify other identities. for example, suppose you wish to know if sin o sec o cot o = 1 is an identity. To find out, simplify the expression on the left side of the equation by using the identities and definitions. sin o sec o cot o = 1 sin o . 1/cos o . 1/tan o = 1 (sec o = 1/cos 0 and cot o = 1/tan o) sin o / cos o . 1/tan o = 1 (multiply sin o and 1/cos o) tan o . 1/tan o = 1 (tan o = sin o/cos o) 1 = 1 (tan o/tan o = 1) Thus, sin o sec o cot o = 1 is an identity. in a way, verifying an identity is like checking the solution to an equation. You do not know if the expression on each side are equal. That is waht you are trying to verify. so, you must simplify one or both sides of the sentence separately until they are the same. (Often it is easier to work with only one side of the sentence. you may choose either side.)
The following suggestions are helpful in verifying trigonometric identities. Study the example to see how these suggestions can be used to verify an identity.
Start with the more complicated side of the equation. Transform the expression into the form of the simpler side. Work with each side of the equation at the same time. Transform each expression separately into the same form. Substitute one or more basic trigonometric identities to simplify the expression. Multiply both numerator and denominator by the same trigonometric expression. There is often more than one way to verify an identity. Remember that verifying an identity is not the same as solving an equation. An identity is true for all values of the variable except those values for which either side of the equation is undefined.
Question: What is the difference between verifying an identity and solving an equation? Answer: An identity is true for all values of the variable except those values for which either side of the equation is undefined, while solving an equation finds specific values where the equation holds true.
Question: Which trigonometric identity was used to simplify "sec(θ)" in the example? Answer: sec(θ) = 1/cos(θ) | 677.169 | 1 |
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Tag Archives: geometry
Post navigationI taught another three-week stained-glass mini-course this year. After my students learn the basic technique of copper-foil stained glass windows, they research a math topic, write a paper on it, and illustrate it with a window of their own design. Topics this year included systems of inequalities, the Fibonacci Sequence, corresponding angles formed by two lines and a transversal, the Four-Color Theorem, and the Pythagorean Theorem among others.
As I was flying home from the 2012 NAIS Annual Conference in Seattle, I looked out airplane's window and saw these patterns in the farmland below. I'm going to figure out a way to use them in my geometry classes. We just finished learning about areas of polygons and circles, so I'm sure these pictures can spark some interesting questions and investigations.
By the way, WCYDWT stands for "What Can You Do With This", a teaching technique pioneered and championed by Dan Meyer. There's a permanent link to his blog at the bottom of my home page.
I'm heading to Seattle this week for the National Association of Independent Schools' annual conference, so I am preparing screencasts for all the classes I'll be absent from. This one is for my geometry students – we just wrapped up areas of polygons and circles, so it's time to add another dimension! Prisms are the simplest solids to work with, so that's how I introduce surface area and volume. All the dirty details are included in the screencast below:
Question: What is the permanent link at the bottom of the author's home page? Answer: A link to Dan Meyer's blog
Question: What did the author's students create after learning the basic technique of copper-foil stained glass windows? Answer: They researched a math topic, wrote a paper on it, and illustrated it with a window of their own design | 677.169 | 1 |
Drag point B to the first quadrant.
Drag the dashed vertical line to the right of the origin.
Similar to side a, we would like to have a vertical segment the
same length as side b. To do this, rotate side b 90°
. Select the origin and Mark Center from the Transform
menu.
Select both side b and its endpoints. Rotate the segment 90°
by choosing Rotate from the Transform menu.
Construct a dashed perpendicular line to the rotated segment through
its endpoint.
Construct the point of intersection between this dashed line and
the vertical dashed line. Change its color to red. Hide both the
rotated segment and its endpoint.
Select the red point and Trace Point under the Display
menu.
Now we are ready to animate. Select point B and the circle and
blue point and the segment along the x-axis.
Choose Action Button under the Edit menu, and then
Animation.
In the Path Match window, make travel selections for your
objects as follows: Point X moves once along segment slowly; Point
B moves once along circle slowly.
To close the Path Match window, click on the Animate
button.
An Animate button will appear in your sketch window. Rename
the button to distinguish it from the other Animate button in your
sketch window.
Double click on the movement button to reset the points; and then
double click on your new Animate button.
Describe your observations
when the button was activated.
Qualitatively and quantitatively
compare and contrast the sine and cosine functions.
For what angle measurements
are the sine function positive? Negative? For what angle measurements
are the cosine function positive? Negative? What are the roots of each
of the functions? How could you find the rotts of the functions mathematically?
For what angle measurement
does the sine function equal the cosine function?
Part 4:
Both the red and green
points are traced when you click on either of your Animate buttons
resulting in both functions being graphed simultaneously. What
was the difference between the Animate buttons? How could you have only
one of the functions to be graphed?
The Geometerís Sketchpad
has a type of Animation button which allows you to show and hide
objects in your sketch window. Create Hide/Show buttons to hide
the traces for each of the trigonometric functions.
To create
a Hide/Show button:
Select the point you wish to hide.
Choose Action Button under the Edit menu, and then
Hide/Show.
Twobuttons will appear in your sketch window. Rename the
buttons with more descriptive names.
Click on the Hide
button you have created for the sine function. Double click on the Animate
button. What changed in your sketch window? Experiment with the other
Hide/Show buttons.
What are the advantages
and disadvantages to the Hide/Show buttons?
Extensions:
Why was the unit
circle chosen to construct the sine and cosine curves?
Using your unit circle,
construct the graphs of other trigonometric functions.
Conjecture what the graph
Question: What is the final step before starting the animation? Answer: Rename the Animate button.
Question: Which menu is used to select the 'Mark Center' option? Answer: Transform menu. | 677.169 | 1 |
Quadrilateral Rap A rap to teach you about quadrilaterals. QUADRILATERAL RAP LYRICS Quadrilateral is a four sided polygon Let's draw some up so pick up yo crayons We got one side, two sides, three sides, four Let's start out with a square and then well talk more More, more Talk more More, more All squares have four All the angles are congruent so there known as equal angular You might even hear a square referred to as rectangular Now we got that covered we can move on to the kite Theres so much to say bout this shape we could be here till the night To the night, night To the night To the night, night Yo my name is Black Knight A kite has two pairs of adjacent and congruent sides If anybody disagrees you tell them they just lied Every convex kite seems to have an inscribed circle And if you dont believe me, man you crazy like Steve Urkle Urkle, Urkle Steve Urkle Urkle, Urkle Inscribed circle A trapezoid is the next shape on our list If you don't listen yo your gunna get dissed They got one pair of opposite parallel sides The other two are legs which will always collide Lide, lide Collide Lide, lide Got rules to abide Diagonals in a parallelogram bisect each other If you dont believe me go ahead and ask yo mother Pairs of opposite angles are also congruent If you listen to these properties youll know it fluent Fluent, fluent Fluent, fluent Opposite angles are also congruent Yo look at the time; Ive reached the end of my rap Thank you, thank you no need to clap See you all later hope you have ...
Guide to Collecting Rare Books : Collecting Autographed and Inscribed Books How much does an autograph increase the value of a book? Learn tips for collecting autographed and inscribed books in this free book collecting video from a longtime bookstore buyer. Expert: Erik Bosee Contact: Bio: Erik Bosse and his family have operated the Aldredge Bookstore in Dallas, Texas for most of its sixty year existence. Erik has bought, sold, and appraised antiquarian books for over 25 years. Filmmaker: Erik Bosse
Question: Which shape has all its angles equal? Answer: Square | 677.169 | 1 |
Short proofs for Pythagorean theorem (Notes in geometry, I). (English)
Int. Math. Forum 5, No. 65-68, 3273-3282 (2010).
Summary: The Pythagorean Theorem can be introduced to students during the middle school years. This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pythagorean Theorem. Students need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials. Through the use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than just $a^2 = b^2 + c^2$ and plugging numbers into the formula.
Question: According to the text, what is one reason why the Pythagorean Theorem is important in high school? Answer: It becomes increasingly important during the high school years. | 677.169 | 1 |
With great encouragement, we should be allowing them time to figure out how they can solve this problem. This is one of the great parts of RS, it teaches children to think and not rely on "set-up" problems. Of course, for those of us that were taught math in the usual manner this will scare us because we are used to being told how to do everything in a formulaic way, so we can pass the test. Here at RS, the idea is to teach to the child to think outside of what they are used to and it will help them to think when they come across new math concepts.
So be prepared this happens often
To answer your question...The 30 degrees should be correct if you are resting it on your T-square, and have your paper taped to the drawing board.
You should have one vertical line going from the top to the center of your circle, then a second 30* angle from the center to the circumference, then a third 30* angle by flipping the triangle and having it go from the center to the circumference. He might do it slightly different, but the idea is the same and it should be fine.
He should be seeing the pattterns in dividing a triangle and a circle in thirds
Question: What is one of the main goals of the teaching method described in the text? Answer: To teach children to think and not rely on formulaic solutions. | 677.169 | 1 |
Hi - I am having a hard time understanding how you know that the two subtriangles, as well as the entire triangle, are each 30 - 60 - 90 triangles "from the information given" as it says in the answer in the back of the book. I know that each has a 90 degree angle in it, but I don't understand how we know that the 90 degree angle at C is split into a 30 degree angle and a 60 degree angle, or how to come to this conclusion based on any other information from the question. Please help!
Question: What is the type of triangle mentioned in the text? Answer: 30-60-90 triangle | 677.169 | 1 |
math In any triangle the sum of the measures of the angles is 180. In triangle ABC, A is three times as large as B and also 16 larger than C. Find the measures of each angle. I also would like to know how to solve these problems: In isosceles trapezoid ABCD, the longer base, AB is ...
Math I need help on two problems. How would you go about solving: 1. 3(a-5) + 19= -2 2. -9 - 3(2q-1)= -18
History My assignment is to write an editorial for the London Telegraph(Britain).I have to write as if it was the summer of 1914 when World War One began and state what i think are the real causes of the World War One through the eyes of Britain. Could you give me some guidance or use...
Biology Why would an Arctic ecosystem be more fragile than a southern forest ecosystem?
Biology Hi, I am studying energy movements in ecosystems. Could you help me with this question: What type of food would be consumed by a secondaryconsumer in the third trophic level of a food chain.
Biology I am not sure how to label each of the following plants,animals and insects with one of these names: producer, herbivore, carnivore, detrius and decomposer. 1.Heron 2.Grasses 3.Adult frog 4.Grasshopper 5.Dead Algae 6.Tadpole 7. Water Boatman 8.Yellow Perch
Science How is it that frogs are parts of two very different food chains?
English Hi, i need to write a very formal essay and was wondering if you can give me some pointers about essay writing.
Question: What type of equation is the first math problem (3(a-5) + 19= -2)? Answer: Linear equation
Question: What is the relationship between angles A, B, and C in triangle ABC? Answer: A is three times as large as B and 16 degrees larger than C | 677.169 | 1 |
An icosahedron is a polyhedron with 20 faces. A regular icosahedron
has triangular faces each of which is an equilateral triangle.
The word icos comes from the Greek for twenty.
In geometry, an icosahedron is any polyhedron having 20 faces,
but usually a regular icosahedron is implied, which has equilateral
triangles as faces. The regular icosahedron is one of the
five Platonic solids. It is a convex regular polyhedron composed
of twenty triangular faces, with five meeting at each of the
twelve vertices. It has 30 edges and 12 vertices. Its dual
polyhedron is the dodecahedron.
Question: What shape are the faces of a regular icosahedron? Answer: Equilateral triangles | 677.169 | 1 |
Pre-Calculus: Adding Vectors & Multiplying Scalars Professor Burger shows you how to add and subtract vectors and use scalar multiplication to elongate or shrink vectors while maintaining their direction angle. The magnitude of a vector can be altered with scalar multiplication. A scalar is simply a number (positive or negative or a fraction) used to multiply a vector by, with the vector keeping its same direction and changing magnitude. Vectors can also be added and subtracted by simply adding or subtracting the components. It is also simple to find the answer graphically by creating a parallelogram with the two vectors, which Professor Burger demonstrates. Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more. Founded in 1997, Thinkwell has succeeded in creating a "next-generation" textbook, one that helps ...
Newton's Laws and vectors Using vectors to determine the horizontal acceleration when force is applied at an angle.
Game Development Tutorial - 17 - Unit Vector What unit vectors are and how to find them with python and pygame.Vectors - The Dot Product Vectors - The Dot Product. I show how to compute the dot product of two vectors, along with some useful theorems and results involving dot products. 3 complete examples are shown. For more free math videos, visit
"Congratulations to Vectors Fellow Sharon Daniel and Co-Creative Director, Erik Loyer! Vectors was also featured on Rhizome and on ReBlog, the Eyebeam blog, in February" — Vectors Journal Blog: Announcements,
"This summer a lot of opportunities await designers all over the world. To help them in their art work I made a roundup of 30 useful free summer vectors on the Web. Check them out!" — 30 Free and Beautiful Summer Vectors | Blog,
Question: What is the purpose of creating a parallelogram with two vectors?
Answer: The purpose is to find the answer graphically by adding or subtracting the vectors. | 677.169 | 1 |
Maths - Points
Here the aim is to work with points in the same terms that we will do with the other geometric elements to be discussed here: lines, planes and volumes. With these other elements we will start with a simple version that goes through the origin and the later add the ability to displace it.
A point which goes through the origin is just the origin, it does not require any information to define its direction, if its associated with a value then its a scalar quantity.
To define a point other than the origin, we can displace it with a vector which is a quantity with magnetude and direction.
It may seem over pedantic to make a distinction between a point and a vector in this way, but the distinction is real and it will be useful later as we will want points and displacements to behave slightly differently.
Symmetry
Symmetry is an important topic for maths and physics.
Symmetry is important for many branches of mathematics including geometry (see this page) and group theory (see this page). Its importance can become apparent in unexpected places, for example, solving quintic equations.
We say that an object is symmetric, with respect to a given mathematical operation, if this operation does not change the object.
Nothers Theorem (discussed further on this page) says that, for every symmetry exhibited by a physical law,
there is a corresponding observable quantity that is conserved. Virtually every theory, including relativity and quantum physics is based on symmetry principles.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.
Applied Geometry for Computer Graphics...
Question: What is the aim of the text regarding points? Answer: The aim of the text is to work with points in the same terms that it will do with other geometric elements like lines, planes, and volumes. | 677.169 | 1 |
4 Answers
Similar to Harald's proof, draw in a radius from the center of the circle to each point where the line intersects the circle. Now draw a perpendicular segment from the center to a point C on the line. Assuming we have more than one point of intersection, we have multiple right triangles which are congruent due to the HL theorem. Clearly we can't have a third point of intersection because there cannot be 3 distinct points along the line equidistant from C.
Or a more geometric proof: If a circle intersects a line in $A$ and $B$, the center of the circle lies on the center normal of the line segment $AB$. If there is a third intersection point $C$, the center of the circle must also lie on the center normal of $BC$. But these two center normals are distinct parallel lines, and cannot have point in common.
Take any 3 distinct points on a circle and notice that each angle of the triangle formed by those 3 points is higher than 0 and smaller than 180 degrees. Any of the angles formed by 3 distinct points on a line (degenerate triangle) takes a value of either 0 degrees or 180 degrees.
Question: Which theorem is used to prove that the right triangles formed by the line, the radius, and the perpendicular segment from the center are congruent? Answer: The Hypotenuse-Leg (HL) theorem. | 677.169 | 1 |
The Hand is one of several symbols sometimes used to designate the conclusion of a mathematical proof by contradiction. In this setting, it is usually drawn horizontally, with little or no adornment, or some minor variant thereof. While less compact than some of the alternatives (such as the blitz or crosshatch), The Hand has the advantage of being visually suggestive of the core concept of this method of reasoning: The conflicting conclusions are represented by the two arrows, and in order to avoid an impasse, one set of assumptions must give way.
The method of proof by contradiction has also been used to show that for any non-degenerate right triangle, the length of the hypotenuse is less than the sum of the lengths of the two remaining sides. The proof relies on the Pythagorean theorem.
We still do not know whether or not our three-dimensional space satisfies the axioms of Euclid's solid geometry, but we do know that we can't use light rays as our straight lines in this geometry. One of the crucial discoveries of physics is that light rays are deflected as they pass near a very massive object. Thus a ray might bend as it passed a star, and the bend would alter the angle sum of a triangle of light rays. That does not mean that our geometry is a non-Euclidean three-dimensional geometry, but it does mean that we have to be careful in trying to apply such a geometry to the study of light rays traveling interstellar distances.
Quoting: Fascinating Timing
Quoting: A Muse Me
Is it really, you call upon the Caticorn and she delivers, lol. The Eris in all things ;)
Question: Is it known whether our three-dimensional space satisfies Euclid's solid geometry axioms? Answer: We do not know for certain. | 677.169 | 1 |
How would I go about calculating the angles you would have to rotate $[0,1,0]$ through the $x$, $y$ and $z$ (although I understand there would only be two angles) axes in order to produce this direction vector? I know how to use rotation matrices and assume they have something to do with it, but I can't make the mental step required to link it all together.
Thanks this is exactly what I was looking for. When I come up with the final solution I'll post it up in an edit of the question, but I reckon this constitutes enough information for a correct answer. – Nick UdellApr 6 '12 at 10:03
Question: What is the user's name according to the text? Answer: Nick Udell | 677.169 | 1 |
Definitions
GNU Webster's 1913
adj.(Math.) the proportion or ratio of squares. Thus, in geometrical proportion, the first term to the third is said to be in a duplicate ratio of the first to the second, or as its square is to the square of the second. Thus, in 2, 4, 8, 16, the ratio of 2 to 8 is a duplicate of that of 2 to 4, or as the square of 2 is to the square of 4.
Question: What does 'geometrical proportion' refer to in this context? Answer: It refers to a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. | 677.169 | 1 |
Re: Circles: Chords, Radii, and Arcs
Yes for 7, that is what I did.
I am not following 8, where did you put COkay, I think Bob has somethingIsn't 8) 1^2 + 1^2 = c^8. If I drew a line segment from A to C, and the radius of circle M was 1, what would line segment AC be? A sqrt 7 B sqrt 2 C sqrt 8 D sqrt 13 E sqrt 4 F sqrt 3
Aren't they 1 not 2From your original drawing AM is 1 and MC is 1 because they are radii. AC is the hypotenuse because it is opposite the right angle. B should be correct. Why did youYou do not need the diameter
Question: Are the radii AM and MC equal? Answer: Yes | 677.169 | 1 |
Ruler
A ruler is an instrument used in geometry to measure short/medium distances and/or to rule straight lines. Strictly speaking, the ruler is the instrument used to rule and calibrated stick for measurement is called a measure. However, common usage is that a ruler is calibrated so that it can measure, creating ambiguity in what a ruler is allowed to do in Ruler-and-compass constructions.
For instance, a ruler with measurement capability (e.g. its own length) can be used for angle trisection. This is resolved by referring to an instrument that can only rule as a straightedge.
Practical rulers have distance markings along their edges.
How these distance markings are applied and calibrated should be described here, including a history of old methods
Question: What is the difference between a ruler and a straightedge? Answer: A ruler is calibrated and can measure distances, while a straightedge is only used for drawing straight lines and does not have measurement markings. | 677.169 | 1 |
2 pairs of proportional sides and congruent angles between them 3. AA Similarity Theorem
2 pairs of congruent angles
Slide 20:
1. PPP Similarity Theorem
3 pairs of proportional sides ABC DFE
Slide 21:
Slide 22:
The PAP Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need. Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.
Slide 23:
Slide 24:
It is possible for two triangles to be similar when they have 2 pairs of angles given but only one of those given pairs are congruent. mT = mX mS = 180- (34 + 87) mS = 180- 121 mS = 59 mS = mZ TSU XZY
Question: Can two triangles be similar if they have two pairs of angles given, but only one of those given pairs are congruent? Answer: Yes | 677.169 | 1 |
Well, I think the title already explains my question. Given a sphere and an ordered sequence of inner angles ($\alpha$, $\beta$, $\gamma$, $\delta$) how many spherical quadrangles do there exist that have that sequence as angles and the added property that three of the edges need to have the same size and the fourth edge needs to have a different size.?
I was told that this on the sphere there might be several quadrangles with quite different appearances, but I can't find any reference explaining this in more detail or bounding the number of possible quadrangles.
You can assume that the angles satisfy the conditions necessary to be the angles of a spherical quadrangle.
2 Answers
The answer for the plane is wrong. You can move any vertex and the two sides incident at it while keeping the opposite vertex and the two sides incident at it fixed, then extend or clip all sides to meet in two more vertices. The result isn't a scaled version of the original quadrangle. This is easiest to imagine for a rectangle or a parallelogram, where you can simply extend two parallel sides.
The same is true for at least some sequences of angles on the sphere. For instance, take two meridians and the equator, which form two right angles, and connect two points at equal latitude on the meridians by a great circle. Now if you move the meridians apart symmetrically, they still form right angles with the equator. The angle they form with the great circle changes, but you can compensate for that by connecting them at a different latitude with a different great circle.
@Will: Sorry, what nonsense; somehow it had slipped my mind that the sides should be geodesics. I deleted that.
–
jorikiFeb 23 '12 at 20:36
1
... and replaced it with a hopefully more sensible example.
–
jorikiFeb 23 '12 at 20:42
Thanks, I updated my question to contain a more correct assessment of the plane case.
I was wondering whether you could limit the number of quadrangles if you add the property that three of the edges need to be equal and one different from the other three.
–
nvcleempFeb 24 '12 at 9:35
@nvcleemp: I think "up to scaling of opposite edges" understates the freedom in constructing quadrangles in the plane. That was just one example that I gave that's easy to visualize; the process that I described is more general than that, and I don't see how it could be described as "scaling" -- certainly not only of opposite edges. I think the statement that there's "a unique quadrangle which has those angles" is simply wrong. On your other question: Do you mean in the plane? In that case, if three of the edge have to be equal, that leaves only one quadrangle up to similarity.
–
jorikiFeb 24 '12 at 11:07
There is no bound on the sphere either. Fix an angle $\alpha > 0.$ For any $\beta > 0,$ Construct the triangle with angles
Question: What is the user's question to joriki about limiting the number of quadrangles on the sphere? Answer: I was wondering whether you could limit the number of quadrangles if you add the property that three of the edges need to be equal and one different from the other three. | 677.169 | 1 |
We can extend our table of sines and cosines of common angles to tangents. You don't have to remember all this information if you can just remember the ratios of the sides of a 45°-45°-90° triangle and a 30°-60°-90° triangle. The ratios are the values of the trig functions.
Note that the tangent of a right angle is listed as infinity. That's because as the angle grows toward 90°, it's tangent grows without bound. It may be better to say that the tangent of 90° is undefined since, using the circle definition, the ray out from the origin at 90° never meets the tangent line.
Angle
Degrees
Radians
cosine
sine
tangent
90°
π/2
0
1
infinity
60°
π/3
1/2
√3 / 2
√3
45°
π/4
√2 / 2
√2 / 2
1
30°
π/6
√3 / 2
1/2
1/√3
0°
0
1
0
0
Exercises
29. In a right triangle a = 30 yards and tan A = 2. Find b and c.
49. cos t = 2 tan t. Find the value of sin t.
Note: In the following problems distance means horizontal distance, unless otherwise stated; the height of an object means its height above the horizontal plane through the point of observation. The height of the observer's eye is not to be taken into account unless specially mentioned. In problems involving the shadow of an object the shadow is supposed to fall on the horizontal plane through the base of the object, unless otherwise stated.
151. The angle of elevation of a tree 250 feet distant is 16° 13'. Find the height.
151. Remember that the tangent of an angle in a right triangle is the opposite side divided by the adjacent side. You know the adjacent side (the distance to the tree), and you know the angle (the angle of elevation), so you can use tangents to find the height of the tree.
152. You know the angle (again, the angle of elevation) and the adjacent side (the distance to the steeple), so use tangents to find the opposite side.
153. Using the angle and the opposite side, use tangent to find the adjacent side.
154. Same hint as in 153.
159. Same hint as in 152.
160. Same hint as in 153.
164. Same hint as in 153.
165. Same hint as in 152.
Question: What is the tangent value of a 45° angle? Answer: 1
Question: In a right triangle with a = 30 yards and tan A = 2, what is the length of side b? Answer: 15 yards | 677.169 | 1 |
Hi Bob, My teacher said everything is correct, but for number 1 she wants me to use trigonometry to solve it. Here is the question again with my original answer:---
What equation (or method) am I supposed to use exactly?
bob bundy
2013-08-13 16:31:45
Yes, that will do nicely.
Bob
demha
2013-08-13 12:07:47
So do I do this: cos(60) = g/12 .5 = g/12 g = 12 x .5 g = 6
So the ladder is 6 feet away from the wall, then we do: 6 - 3 = 3. The base of the ladder from the bottom of the fence is 3 ft.
Is this method correct?
bob bundy
2013-08-13 07:27:13
Hi demha
Well done for sorting out Q3.
For Q8, I think the questioner intends that the ladder just scrapes the top of the fence. (Actually now I think about it, there is no need to assume this. You can do the question anyway.) I have copied your diagram but made the fence taller and labelled the points.
Too big. You've muddled up which side is which in the formula. 4. Q. A 30-60-90 triangle has a hypotenuse of 16 and a short side of 8. Use special right angle formulas to find the third side. Show your work. Does your answer match what you got on number 3?
A. 8 (sqrt3) = 13.856
Correct. 5. Q. A 45-45-90 triangle has a leg of 4[sqrt(2)]. What is the hypotenuse? Show your work.
A. 4 (sqrt2) x (sqrt2) = 4 (sqrt4) = 8
Correct. 6. Q. A right triangle has legs of 4 and 5. What is the hypotenuse? Show your work.
correct. 8 Have you made a diagram? Using 12 and the special formula you can work out base of ladder to wall. Then it's easy.
Bob
demha
2013-08-13 02:02:26
This is a review I need to do. It will involve Triangles, Sohcahtoa, Pythagorean Theorem and the Special Right Triangles (30 - 60 - 90 and 45 - 45 - 90). Please help me understand them and check if they are correct: (NOTE FOR #3 and #4, there must be something wrong. I thing I am doing something wrong in #3. Help would be appreciated! I would like to learn a method to find the answer if mine is wrong.)2.
Question: Which theorem is used to find the hypotenuse of the right triangle with legs of 4 and 5? Answer: Pythagorean theorem
Question: What is the height of the fence in Bob's diagram for question 8? Answer: The height of the fence is not explicitly stated in the text, but it is mentioned that Bob made the fence taller in his diagram. | 677.169 | 1 |
Common Core Standards: Math
Math.G-C.4
4. Construct a tangent line from a point outside a given circle to the circle.
It's construction time, so tell your students to put on their hard hats. Actually, don't. You'll only get groans and eye rolls.
Students should already know that constructions involve straightedges and compasses rather than jackhammers and drills. Well, it depends on what you mean by drills. Hopefully they've become adept enough in using these instruments because they'll have to put them both to use.
If students don't already know the properties of circles and tangents before this construction, they should take away a few main points from it:
Tangents drawn to a circle are perpendicular to the circle's radius at the point of tangency.
Two tangents drawn to a circle from the same point outside the circle are equal. You can have students do this construction and measure the segments from the point of tangency to the shared point.
Two tangents drawn to a circle from the same point outside the circle make an angle that, when bisected, includes the circles center. Students can construct the angle bisector and see for themselves.
Tangents to a circle at either end of a diameter are parallel.
Given a circle with center B and a point A outside the circle, the construction of a tangent to ⊙B that goes through A is relatively simple. What's important is that students also understand the properties of circles and their tangents in order to make these constructions
Question: What happens when two tangents are drawn to a circle from the same point outside the circle? Answer: They are equal in length | 677.169 | 1 |
Definition 3
The line AB is cut in extreme and mean ratio at C since AB : AC = AC : CB.
A construction to cut a line in this manner first appeared in Book II, proposition II.11. Of course that was before ratios were defined, and there an equivalent condition was stated in terms of rectangles, namely, that the square on AC equal the rectangle AB by BC. That construction was later used in Book IV in order to construct regular pentagons and 15-sided polygons (propositions IV.10 through 12 and 16).
Now that the theory of ratios and proportions has been developed, it is time to define this section as a ratio, rather than using rectangles. An alternate construction is given in proposition VI.30.
Question: What is the main reason given for defining this section as a ratio now? Answer: Because the theory of ratios and proportions has been developed | 677.169 | 1 |
Precalculus
Topics
The Hyperbola
A hyperbola is a type of conic section that is formed by intersecting a cone with a plane, resulting in two parabolic shaped pieces that open either up and down or right and left. Similar to a parabola, the hyperbola pieces have vertices and are asymptotic. The hyperbola is the least common of the conic sections.
Question: What is the name of the curve that a hyperbola approaches but never reaches? Answer: Asymptote | 677.169 | 1 |
The Coordinate Plane
A coordinate plane is the rectangular plane formed with two number lines: one is vertical number line and the other one is the horizontal number line. In the horizontal number line, at left side, negative numbers are placed and at the right hand side, the positive numbers are placed.
In case of vertical axis, upper part contains positive numbers and lower part contains negative numbers. The horizontal line represents the x-axis and the vertical line represents the y-axis. There are four quadrants and two axis in a Coordinate Plane. The x-axis is also named as Abscissa and the other name of y-coordinate is Ordinate. Suppose we have to locate a point, then we require a coordinate which is known as ordered pair and these are Set of 'x' values and 'y' values. The first coordinate in the ordered pair known as Domain which is usually the x-coordinate, and the second coordinate which is usually the y-coordinate is known as Range. The coordinate plane is a two-dimensional surface on which lines, points and curves are plotted with the help of ordered pair. Origin is the point where 0 is placed and 0 is equidistant from both horizontal and vertical axis. Two axes divide the plane into four quadrants. As we know that coordinate plane is made with the help of convection, convection are also helpful in defining the quadrants, the first quadrant is at the top right side of the plane and rest are counted in anti-clockwise format.
History of Coordinate Plane
Coordinate plane is a plane which has two axes and four quadrants. Horizontal number lines are along x - axis and vertical number lines are along y - axis. In Coordinate Plane, center Point is known as origin. Its coordinate value is (0, 0). Now, let's have a look on History of Coordinate Plane. Concept of Coordinate plane was given by French mathematician, philosopher and scientist "Rene Descartes". He was a very strong thinker. Rene Descartes gave the concept of analytic Geometry and Cartesian coordinates and Cartesian curve. Coordinate plane was first invented in 1637 by Descartes and Pierre de Fermat. Fermat also worked on three dimensions, but he did not publish his discovery. Both authors used single axis. Invention of coordinate plane system plays an important role in development of Calculus, that was given by Isaac Newton and Gottfried Wilhelm Leibniz. Since times of Descartes, many different coordinate system has been invented, such as, polar coordinate, that was given for plane, spherical and cylindrical coordinate, that was given for three dimensional space.
Figure of coordinate plane is shown below:
Question: Who invented the concept of the coordinate plane? Answer: René Descartes.
Question: Which famous scientists contributed to the development of calculus using the coordinate plane? Answer: Isaac Newton and Gottfried Wilhelm Leibniz.
Question: Who also worked on the concept of the coordinate plane but did not publish his discovery? Answer: Pierre de Fermat. | 677.169 | 1 |
13.Corollary 4-2-2: The acute angles of a right triangle are complementary.
14.Corollary 4-2-3: The measure of each angle of an equiangular triangle is 60°.
18.Corollary 4-8-3: If a triangle is equilateral, then it is equiangular.
19.Corollary 4-8-4: If a triangle is equiangular, then it is equilateral.
21.Corollary 7-4-3 Two-Transversal Proportionality Corollary: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.
22.Corollary 8-1-2 Geometric Means Corollary: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse
23.Corollary 8-1-3 Geometric Means Corollary: The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
26.Corollary 11-4-2: If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent.
1.Postulate 1-1-1: Through any two points there is exactly one line.
2.Postulate 1-1-2: Through any three noncollinear points there is exactly one plane containing them.
3.Postulate 1-1-3: If two points lie in a plane, then the line containing those points lies in the plane.
4.Postulate 1-1-4: If two lines intersect, then they intersect in exactly one point.
5.Postulate 1-1-5: If two planes intersect, then they intersect in exactly one line.
6.Postulate 1-2-1 Ruler Postulate: The points on a line can be put in a one-to-one correspondence with the real numbers.
7.Postulate 1-2-2 Segment Addition Postulate: If B is between A and C, then AB + BC = AC
8.Postulate 1-3-1 Protractor Postulate: Given line AB and a point O on line AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180.
9.Postulate 1-3-2 Angle Addition Postulate: If S is in the interior of angle PQR, then the measure of angle PQS + the measure of angle SQR = the measure of angle PQR.
10.Postulate 3-2-1 Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Question: Which postulate states that through any two points there is exactly one line? Answer: Postulate 1-1-1
Question: Which corollary states that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse? Answer: Corollary 8-1-2
Question: Which postulate allows us to measure angles using a protractor? Answer: Postulate 1-3-1 | 677.169 | 1 |
statistics A simple random sample of 50 items from a population with σ 6 resulted in a sample mean of 32. a. Provide a 90% confidence interval for the population mean
math in triangle abc ad is bisector of angle a and angle b is twice of angle c prove that angle bac is equal to 72
Physics Thanks, for the help Elena. For me this answer is also not working its saying that its wrong. :/
Physics Four people sit in a car. The masses of the people are 41 kg, 47 kg, 53 kg, and 55 kg. The car's mass is 1020 kg. When the car drives over a bump, its springs cause an oscillation with a frequency of 1.00 Hz. What would the frequency be if only the 41-kg person were presen...
Question: What is the measure of angle BAC in triangle ABC? Answer: 72 degrees | 677.169 | 1 |
Geometry Teacher Resources
Find Geometry educational ideas and activities
Title
Resource Type
Views
Grade
Rating explore geometry using a Rubik's Cube. In this 2-D and 3-D shapes lesson plan, students use the Rubik's Cube to find the center, edge and corner pieces. Students then find the dimensions of the Rubik's Cube and read the solution guide. After reading students demonstrate methods and algorithms.
How can we identify shapes in the Universe? High schoolers will compare and contrast elliptic and hyperbolic geometry. They will also explore one possible way to measure the curvature of the Universe, namely, by measuring the sum of the angles in a triangle. Resource links are included.
Tenth graders explore non-Euclidean geometry. In this geometry lesson plan, 10th graders investigate "taxicab geometry." Students investigate the taxicab distance formula and make comparisons between a circle in Euclidean geometry and a circle in taxicab geometry.
In this geometry worksheet, 10th graders review geometric concept including angles, polygons, quadrilaterals, trig and circles. They take a comprehensive exam covering topics on the Regents exam. There are 32 pages in this booklet.
Tenth graders investigate the early history of geometry. In this geometry lesson, 10th graders investigate translations, rotations, and reflections. They also solve problems with line of symmetry and rotational symmetry while reviewing important symmetry theorem.
Students create a quilt square for a class quilt using at least three, two-dimensional geometric figures. They research and write a brief description of at least two different quilt patterns that they find. Pupils discuss that quilts are not only a part of America's heritage, and relate it to math and geometry. Students are introduced to a variety of geometric figures during a unit of geometry.
Origami is an excellent way to combine Japanese culture, art, and geometric shapes into one engaging lesson plan! Scholars begin by listening to the story Sadako and the Thousand Paper Cranes and learn the origin of the word origami, geometry, and symmetry. Learners watch a short intstructional video depicting a simple origami project and locate shapes they recognize. Next, they practice origami themselves using the intructional prompts. Consider doing this yourself on a document camera, or having an experienced artist come speak to the kids about this practice. As learners fold, they pay attention to familiar shapes. There is a worksheet to solidify geometric concepts, and the lesson plan suggests taking pictures of the origami stages to create a PhotoStory presentation.
The University of New York Regents High School Exam for geometry from August 2009 is comprehensive in scope with 38 questions over 22 pages. Geometers can assess their mastery of core content with a combination of multiple choice and constructed response questions. A reference sheet with relevant formulas is included.
Learners explore transformations. For this middle school geometry lesson, students analyze movement of a shape from one place to another on a plane. The lesson requires the use of the geoboard application of the TI-73.
Question: What is one way high schoolers can measure the curvature of the Universe according to the second lesson plan? Answer: By measuring the sum of the angles in a triangle.
Question: What is the total number of pages in the geometry worksheet for 10th graders? Answer: 32 pages | 677.169 | 1 |
A parabola is the set of points equidistant from a line, called
the directrix, and a fixed point, called the focus. Assume the
focus is not on the line. Construct a
parabola given a fixed point for the focus and a line (segment)
for the directrix.
Here is a Geometer's Sketchpad script that shows the construction
of the parabola. The first two points
that you put in the plane represent the endpoints of the directrix.
The third point represents the focus. If you move the point that
appears on the directrix, you can trace out a parabola. Describe
what happens when you move the focus away from the directrix.
What happens when the focus is very close to the directrix? Make
some conjecture about what will happen if the focus is placed
on the directrix.
You can animate this parabola by clicking here
and then double clicking on the animate button. When you want
to stop the animation, simply click the mouse again.
Let us look at this animation again and trace the tangent line
at the constructed point. Click here
to see this animation.
Use the locus command to generated the parabola from a constructed
point or the tangent line at that point. To construct the locus
of an object, select that object, and then select some point constructed
on a path which, when it moves along its path, defines the position
of your selected object.
Question: What would happen if the focus were placed on the directrix? Answer: If the focus were placed on the directrix, the parabola would degenerate into a single point, as all points on the directrix are equidistant from the focus. | 677.169 | 1 |
Begin with a point O. Extend a line in opposite directions to points A and B a distance r from O and 2r apart. Mark the centre O. Draw a line COD of length 2r, centred on O and orthogonal to AB. Join the ends to form a square ACBD. Draw a line EOF of the same length and centred on 'O', orthogonal to AB and CD (i.e. upwards and downwards). Join the ends to the square to form a regular octahedron. And so on. These are the cross polytopes.
Point
Line segment
Square
Octahedron
16-cell
History of discovery
Convex polygons and polyhedra
The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek mathematicians. The five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (ca. 417 B.C. – 369 B.C.) described all five. Later, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.
Higher-dimensional polytopes
A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. It rotates about the zw plane.
It was not until the 19th century that a Swiss mathematician, Ludwig Schläfli, examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in (Schläfli, 1901), six years posthumously, although parts of it were published in 1855 and 1858 (Schläfli, 1855), (Schläfli, 1858). Interestingly, between 1880 and 1900,
Schläfli's results were rediscovered independently by at least nine
other mathematicians — see (Coxeter, 1948, pp143–144) for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by Hoppe in 1882, and first used in English by Mrs. Stott some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).
Question: When were Schläfli's results first published in full? Answer: 1901, six years posthumously
Question: Who were the earliest known mathematicians to study regular polygons and polyhedra? Answer: Ancient Greek mathematicians
Question: In which century were regular polytopes in higher dimensions first examined and characterized? Answer: 19th century | 677.169 | 1 |
The curriculum is aimed at children who are learning about shapes for the first time,whereas you are adults looking at shapes from a different perspective. You have alreadymade certain generalisations about shapes which young learners may not yet have made.Their fresh view on shapes enables them to distinguish shapes in different ways. You as ateacher can tap into this fresh and clear perspective on shapes. Why do you think it is appropriate for young learners tostudy space shapes like balls, cones and boxes before they study circles, triangles and rectangles? ReflectionIn abstract geometry we work with points all of the time. We need to be able to drawthem, and to name them clearly. We draw dots to represent points and we use capitalletters to name points.Remember that points have no size. It does not matter what size dot you draw, althoughwe usually draw the dots quite small, to help us to focus on the point. P and Q below areboth points. We visualise the point at the centre of the dot we have drawn, no matter thesize of the dot. •Q • P Activity 1.1 1. If points have no size, what difference does the size of the dot make? Activity 2. If you were to join P and Q with a line segment, could you locate some more points on the line between P and Q? 3. How many points can you find between any two other points?The use of capital letters to name points is simply a convention. There are manyconventions (the generally accepted correct form or manner) in mathematics. We need toknow and use these correctly and pass on this knowledge to our learners so that they willbe able to speak and write correctly about the shapes they are dealing with.The Cartesian planeWe dont only need to name points. Sometimes we need to locate points in particularlocations. To do so we use the Cartesian plane. This is a system of two axes drawnperpendicular to each other, named after Descartes, a French mathematician andphilosopher who thought about the idea of placing the two axes perpendicular to eachother to facilitate the location of points in a two-dimensional plane.
When we name points in the Cartesian plane we name them as ordered pairs, also knownas coordinate pairs. The coordinates come from each of the axes and must be given in theorder of x-co-ordinatefirst and then y-co-ordinate. The x-axis is the horizontal axis andthe y-axis is the vertical axis. Look at the following diagram of a Cartesian Plane: Activity 1.2 1. Did you know all of the terminology used in naming the points and axes above? If not, spend a little time studying and absorbing Activity the terms you did not know. They should all be familiar to you. 2. What are the co-ordinates of B and C? Write them up as ordered pairs. 3. What shape do you form if you join points A, B and C with line segments? 4. Could you have formed a different shape if you did not have to join the points with line segments? Experiment on the drawing to see what shapes you can make!
Question: What is the difference between a point and a dot in the context of the text? Answer: A point has no size, while a dot is used to represent it and can have any size.
Question: What are the two axes in the Cartesian plane called? Answer: The x-axis (horizontal) and the y-axis (vertical).
Question: Which shapes are children expected to study before they study circles, triangles, and rectangles? Answer: Balls, cones, and boxes.
Question: What is the shape formed by joining points A, B, and C with line segments in the provided diagram? Answer: A triangle. | 677.169 | 1 |
Plane and space shapesNow look at the drawings of shapes below.We call each of them a geometric figure or shape.A B C DE F G HI J K Activity 1.3 1. Can you name each shape? Write the names next to the letter corresponding to each shape. Activity 2. Look carefully at each shape again. As you do so, think about their characteristics. Write down some of the characteristics that you thought about. Try to think of at least one characteristic per shape.Whether you could name the shape or not and whatever characteristics of the shape youcould give, if you look again you will quickly notice (if you have not already) that someof the shapes are FLAT and some of the shapes protrude into SPACE.This is the first major distinction that we are going to make in terms of geometric figures– some are called PLANE FIGURES (they are flat and lie in a plane or flat surface;examples are B, C, E, F, H, I and Jabove). Others are SPACE FIGURES (they are not flatand protrude from the surface on which they are resting; examples are A, D, G and K
above). Plane and space are separate from the dimension of the shape – they tell uswhether the shape is flat or not.DimensionDimension tells us something else. Let us see how dimensions are defined.If we look even more closely at the shapes we can see that they are not all made in thesame way. Some are solid, some are hollow, some are made of discrete (separate) points;others are made of line segments, curves or surfaces. Refer again to the drawings of shapes above when you read these notes. Note Look at shape B. It is made of points. We call this kind of shape zero- dimensional (0-D). Look at shapes E, F and I. They are made of lines or curves.We call this kind of shape one-dimensional (1-D). Look at shapes C, Hand J. They are made of flat surfaces. They can lie flat in a plane. We call this kind of shape two-dimensional (2-D). Look at shapes A, D, G and K. They protrude into space. They do not sit flat in a plane. We call this kind of shape three-dimensional (3-D).This definition of dimension might be a bit more detailed than definitions of dimensionwhich some of you may already have heard or know. You may only have thought about2-D and 3-D shapes. It is analytical and corresponds closely to the make-up of eachshape. It could also correspond with a very close scrutiny of each shape, such as a childmay subject each shape to, never having seen the shape before and looking at it with newand eager eyes. To teach well we need to be able to see things through the eyes of a child.
Question: What is the purpose of teaching with the perspective of a child, as mentioned in the text? Answer: To teach well, one should be able to see things through the eyes of a child, especially when they are encountering new or complex concepts. | 677.169 | 1 |
Activity 1.4 1. Remember that we distinguishedbetween plane shapes and space shapes.What is the main difference between plane shapes Activity and space shapes? 2. What dimensions could plane shapes be? 3. What dimensions could space shapes be? 4. Here are some sketching exercises for you to try. On a clean sheet of paper sketch and name the following shapes: a zero-dimensional plane shape a one-dimensional plane shape a two-dimensional plane shape a three-dimensional space shape. 5. Sketch any three other shapes of your own (or more if you would like to), and then classify them according to plane or space and dimension.Plane figuresWe now take a closer look at the flat geometric shapes. They are called plane figures (orshapes) because they can be found in a plane (a flat surface). Think of the geometric figures that are studied in the primary school – which of them are flat? Write down the names of all the FLAT shapes of which you can think. Reflection The first plane figure that we define is a curve. What do you think a curve is? Sketch it in your notebook. Do more than one sketch if you have more than one idea of what a curve looks like.
CurvesDid your curve(s) look like any of the following shapes?A B CD E FStudy the above shapes carefully. A, B, C and D are all curves. They are all one-dimensional connected sets of points. They can be drawn without lifting your pen fromthe paper. You may have thought that only A and B are curves and the idea that C and Dare also curves might surprise you. Curves are allowed to be straight! Dont forget this!Several of the plane shapes that we study are made up of curves. We will now have alook at some different kinds of curves. Look at the sketches below. Circle all those that represent single Reflection curves. (Try this without looking at the solutions first!)
A B CD E FG H I J Check your answers and correct them if necessary. The sketches that you should have circled are A, D, E, F, H, I and J. You can draw each of these without lifting your pen. Discussion B and G are made of more than one curve. C is zero dimensional – it is made of discrete points. Examine all of the shapes which we have said are curves. What do they all have in common? Are there also differences between the shapes which are curves? Try to Reflection describe these differences.
Kinds of curvesBecause of the differences which you have just been recording, we can categorise curvesinto different groups, according to the following terms: open/not open (closed); andsimple/not simple (complex).Study the drawings below to see if you can come up with your own definitions of theterms open, closed, simple and complex.All of these are simple closed curves None of these are simple closed curves
Question: What are the dimensions of a one-dimensional plane shape? Answer: One dimension | 677.169 | 1 |
Can we make a generalisation about the relationship between the number of axes of (line) symmetry of a shape and its order of rotational symmetry? Reflection Activity 1.24 Complete the information in the following table.Then study the Activity completed table and see what you can conclude. Figure Number of Angles of Order of lines of rotational symmetry symmetry symmetry Square Equilateral triangle Regular octagon Regular nonagon Rectangle Circle ParallelogramA 1800 rotation in a plane about a point results in what we call point symmetry(i.e. pointsymmetry is a particular type of rotational symmetry.)As with line symmetry, we talk about point symmetry with respect to a point and pointsymmetry for pairs of figures. You must be able to draw and recognise when a shape haspoint symmetry with another shape. The figures below are point symmetrical.
TranslationsWhen a shape has been translated, every point in the shape is moved the same distance.You can involve the learners in point plotting exercises and get them to pull or pushshapes ALONG a line, BELOW or ABOVE a line, and so on, to draw translations. Theshapes in the grid below have been translated.The arrows show the direction of the translation each shape has undergone. Horizontal translations move the shape to the left or right without any upward or downward movement. Vertical translations move the shape up or down without any sideways movement. Oblique translations move the shape to the left or right and upward or downward movement at the same time. The arrow indicating the movement is at an angle to the horizontal. What is the difference between the description of a horizontal/vertical translation and an oblique translation? Reflection Activity 1.25 1. Draw a few shapes on grid paper and then translate them. Record what translations you make the shapes undergo. Activity 2. Shapes can be translated vertically, horizontally and obliquely. On a grid translate a shape in each of these ways. Record the translations made by each shape.
Question: Which shape in the table has 6 lines of rotational symmetry? Answer: Regular hexagon
Question: What is the key difference between line symmetry and point symmetry? Answer: Line symmetry involves reflection over a line, while point symmetry involves reflection over a point. | 677.169 | 1 |
PROBLEMS FOR DECEMBER
Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than January 31, 2002.
Note. The incentre of a triangle is the centre
of the inscribed circle that touches all three sides. A set
is connected if, given two points in the set, it is
possible to trace a continuous path from one to the other
without leaving the set.
121.
Let
n be an integer exceeding 1.
Let
a1,a2,…,an be posive real numbers
and
b1,b2,…,bn be arbitrary real numbers for which
∑i≠jaibj=0.
Prove that
∑i≠jbibj<0.
122.
Determine all functions
f from the real numbers
to the real numbers that satisfy
f(f(x)+y)=f(x2-y)+4f(x)y
for any real numbers
x,
y.
123.
Let
a and
b be the lengths of two opposite
edges of a tetrahedron which are mutually perpendicular and
distant
d apart. Determine the volume of the tetrahedron.
and sketch this set in the complex plane.
(Note: Im and Re refer respectively to the imaginary
and real parts.)
126.
Let
n be a positive integer exceeding
1, and
let
n circles (i.e., circumferences)
of radius 1 be given in the plane such that
no two of them are tangent and the subset of the plane formed
by the union of them is connected. Prove that the number of
points that belong to at least two of these circles is at least
n.
Question: What is the definition of the incentre of a triangle? Answer: The incentre of a triangle is the center of the inscribed circle that touches all three sides. | 677.169 | 1 |
Gail then asked the group how students are going to understand what trigonometry has to do with non-right triangles, as the law of sines will not have been introduced yet. Allen continued to suggest that perhaps the lesson could ask "why doesn't right triangle trigonometry work for all triangles?
Ellie talked about how some students might make side to side ratios of one triangle, while others might make side to side ratios of two similar triangles, and that these are two very different ways to look at the basic proportions. This led to a group discussion about the meaning of these ratios, specifically misconceptions students might have in moving from right triangle trig to non-right triangle trig. The lesson plan for the day before the group's lesson was re-examined and the lesson seemed "traditional" in that right triangle trigonometry would be covered using the mnemonic "SOHCAHTOA" and some practice problems.
The group decided that the big question of how similar triangles and trigonometry are related will be the focus for the lesson study. The group decided that as homework, each member would look into problems that get at this question and arouse the student's curiosity at the same time.
DAY 4 (7/1/05)
Kathy Kanim joined the group for the meeting and described her experience brining lesson study to her district in New Mexico last year. Gail summarized the previous meeting by describing where the group had gotten "stuck" the day before. For example, she reminded the group that Ellie had brought up how students write ratios very differently, and that this will affect the transition from these ratios to proportions to trigonometry.
Claudia relayed how difficult choosing a problem will be and Joyce added that this is a very important big idea. Allen suggested asking the students Why does a non-right triangle not work with the right triangle trig functions? Or How can we apply the right triangle trig function to a non-right triangle? Jeremy said that this would be too hard to answer, especially with students the group doesn't know. Amy continued to suggest the question could be CAN you use right triangle trig functions for any kind of triangle? Jeremy emphasized his previous point, explaining that this is a big idea that might be too hard for any student.
Teachers began sharing specific questions they had found the previous night:
Do the values of trig functions change as a triangle changes? (in terms of similar triangles) (Mary)
A lot has a perimeter of 1200 ft. and a shape of a right triangle. It is going to be divided into two smaller lots in the shape of right triangles such that the area of one is half the other. If a fence is to be constructed to separate the two lots, what angles does it make with the two short sides of the original lot? What are the areas of the two lots? What happens if the original lots' perimeter is doubled or tripled? (Rey)
The group all focused on doing the math of Rey's problem. Teachers worked individually and in pairs and put up answers on the board. Once all members understood the math behind the problem, the group began discussing the math that the problem evokes.
Proportional reasoning
Question: What is the acronym used in the traditional lesson for right triangle trigonometry? Answer: SOHCAHTOA | 677.169 | 1 |
Application of the Pythagorean theorem
Knowledge of right triangle properties
Understanding of similar triangles
Though everyone agreed the problem was a good one in and of itself, they questioned the usefulness of it in addressing the big question of how similarity and trigonometry are related. Gail asked if there are an infinite number of triangles that would work with perimeters other than 1200. Kathy thought it would be just as interesting to look at the triangles with perimeters of 1200 that don't work. Joyce, in looking at the textbook the students would be using, relayed that the chapter on similarity comes at the end of the book and that the students might not have that background going into the lesson. Amy and Aki wondered how this problem would get us to non-right triangles. Mary responded by saying that the same process would be used for non-right triangles to find missing sides or angles (i.e. the law of sines).
Aki suggested posing a problem where at least two solutions involve using similarity and using trig, respectively. He relayed that he thought this might be a good way to connect the two. He continued to say "if you change the area ratio so that the area of one triangle is one-third of the other it becomes a 30-60-90 triangle," which he suggested might provide more access into the problem. Allen drew a diagram on the board to illustrate the new idea. Gail finally said that to get to the trigonometry, however, an angle needs to be given off the bat in some way.
The group re-focused on Ellie's problem, which asked students the following:
You are building a house on an island. There is a utility tower on the other side of the river [creating the island]. The installation cost for electric service is based on the distance from the utility pole. The county surveyors have provided the location where the angle to the utility pole is 90 degrees and another location 30 meters away (along the same line), where the angle to the tower is 48 degrees. Where should I build the house to minimize the distance to the tower? If the installation cost is $1.50 per meter, how much would the installation charge be?
Ellie drew a diagram on the board similar to the following but the group decided the problem would hit trigonometry via indirect measurement, but not similar triangles. A few other teachers gave suggestions, but none of the problems hit both targets. The group decided to re-examine Rey's problem and focus on Aki's editing. Specifically, the group decided more research needed to be done around finding a way to make Rey's problem solvable using similarity or trigonometry.
Question: What is the installation cost per meter for electric service in Ellie's problem? Answer: $1.50
Question: What is the ratio of the areas of the two triangles when Aki suggested changing the area ratio? Answer: 1:3
Question: Who decided to re-examine Rey's problem for further edits? Answer: The group of teachers | 677.169 | 1 |
How do you transform a circle into a doughnut? Transform 2D shapes to make a stack of 3D objects. For example, spin a circle around a lateral axis to form a doughnut-shaped solid. Or you could extrude the circle (like toothpaste) to form a cylinder.
Choose a shape and imagine it being spun around an axis or extruded. Look at a series of 3D blocks and predict which one your 2D shape will make. Picture other transformations in your head and identify the solids produced.
Question: In which direction is the circle spun to form a doughnut-shaped solid? Answer: Around a lateral axis. | 677.169 | 1 |
Problem Solving Unit
Problem 3.1 Triangular and Square Numbers
a. Write triangular numbers up to 100.
b. Write square numbers up to 100.
c. State the relation that exists between the square numbers and the
triangular numbers.
Problem 3.2 The Age
A's age equals B's age minus the cube root of C's
age. B's age equals C's age minus the square root of A's
age, plus 24 years. C's age equals the square root of A's
age plus the cube root of B's age. Find, by trial and error,
the age of:
a. A in years
b. B in years
c. C in years
Problem 3.3 Poppy
Poppy's dad paid her 50 cents for every problem she got right but
penalised her 30 cents for any problem she got wrong. After
attempting 40 problems Poppy received only 80 cents. How many answers did
she get right?
Problem 3.4 Triangular Pyramid
1. Apples are arranged so that they form the shape of an
equilateral triangle. State the number of apples in the following
diagrams.
2. Apples are stacked in a triangular pyramid. Each layer of apples
is in the shape of an equilateral triangle, and the layer at the top has
one apple as shown in the following diagram.
Question: What is the largest triangular number less than 100? Answer: 91
Question: What is the relationship between square numbers and triangular numbers? Answer: Square numbers are always greater than or equal to triangular numbers. | 677.169 | 1 |
Geom2-2HandoutDocument Transcript
Section 2.2 Geogebra Activity<br />This activity will be completed in groups of 2-3. Please follow all directions and collaborate with your group members to complete all questions. <br />One group member should record all answers and information on the ANSWER SHEET, and a different group member should manipulate the program on the netbook. The third group member is responsible for keeping track of the time (you will be given a time limit) and reading these instructions out loud. If you do not have a third group member, delegate the task of reading aloud as you see fit.<br />DECIDE WHO WILL PERFORM EACH TASK AND RECORD THIS ON THE ANSWER SHEET.<br />
Go to This might take a couple of minutes to load. GeoGebra is a free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. It has received several educational software awards in Europe and the USA.
Click on "View" in the Geogebra toolbar (not the web browser's toolbar). Select "Axes" in order to remove the x- and y- axes from the viewer.
Select the line tool.
You will create 2 lines. Simply click once to begin to place the line, and then click the second time to determine the direction of the line. Make sure to leave some distance between the two points you click. The intersection of these lines should be between the points that name the line.
Click the small downward arrow on the construction button. When the menu drops down, select "Intersect Two Objects." You will then click on each line (anywhere on the line EXCEPT the points). This should create a point F at the intersection.
We are almost there! Measure 2 adjacent angles. Select the angle tool. You will click on 3 points to define the angle. THE ORDER IN WHICH YOU SELECT THE POINTS IS VERY IMPORTANT. You must select the points in a clockwise direction with the vertex being the second point you select.
Lastly, select the arrow tool. Click on a point that is not the intersection, and manipulate the angles. Record some of your angle measures on the ANSWER SHEET.
ANSWER SHEET<br />Please list your group members and their jobs here:<br />Once you manipulate the angles, record some of the measures you saw. Record the measures in pairs. For example, if I see an angle of 66.14 degrees, I would see an angle of 114.86 degrees at the same time. Record BOTH of these measures as a pair.<br />1.1.2.2.3.3.4.4.<br />Talk it over with your group, and make some observations about the pairs of angle measures. Look at their numerical relationship as well as how the numbers relate to the picture.<
Question: What should be done to remove the x- and y- axes from the viewer in GeoGebra? Answer: Click on "View" in the GeoGebra toolbar and select "Axes". | 677.169 | 1 |
trigonometry
Plane trigonometry
In many applications of trigonometry the essential problem is the solution of triangles. If enough sides and angles are known, the remaining sides and angles as well as the area can be calculated, and the triangle is then said to be solved. Triangles can be solved by the law of sines and the law of cosines (see the table). To secure symmetry in the writing of these laws, the angles of the triangle are lettered A, B, and C and the lengths of the sides opposite the angles are lettered a, b, and c, respectively. An example of this standardization is shown in the figure.
The law of sines is expressed as an equality involving three sine functions while the law of cosines is an identification of the cosine with an algebraic expression formed from the lengths of sides opposite the corresponding angles. To solve a triangle, all the known values are substituted into equations expressing the laws of sines and cosines, and the equations are solved for the unknown quantities. For example, the law of sines is employed when two angles and a side are known or when two sides and an angle opposite one ... (200 of 6,336 words)
Question: Which law identifies the cosine with an algebraic expression formed from the lengths of sides? Answer: The law of cosines. | 677.169 | 1 |
The answer by @Licson below is a better answer. It returns an angle in $(-180,180]$ unlike the answer below that uses $\arccos$. I don not understand why that answer got a negative vote.
–
copper.hatMay 2 at 5:15
Agree. The arccos answer will return an angle modulo pi radians, which throws away "directional" information about the two points.
–
Trevor AlexanderNov 22 at 23:59
Question: Who provided the second comment in the text? Answer: Trevor Alexander | 677.169 | 1 |
icosidodecahedron
Definitions
from Wiktionary, Creative Commons Attribution/Share-Alike License
n. A semiregular polyhedron with twelve faces that are regular pentagons and twenty that are equilateral triangles.
from The Century Dictionary and Cyclopedia
n. In geometry, a solid of thirty-two faces formed by cutting down the corners of the icosahedron parallel to the faces of the coaxial regular dodecahedron until the new faces just touch at the angles, thus leaving 20 triangular and 12 pentagonal faces. It is one of the thirteen Archimedean solids.
Question: What are the shapes of the faces of an icosidodecahedron according to The Century Dictionary and Cyclopedia? Answer: 20 equilateral triangles and 12 regular pentagons | 677.169 | 1 |
Homework Exam 1, answers, Geometric Algorithms, 2013. Question 1 (3 points). In Chapter 1 we saw how to compute the convex hull of a set P of n points in the plane. The essential ingredient is testing whether a point q lies in a cycle.
Part I: Answer ONE (1) of the following questions 30%. the rate of savings and the "ICOR" in determining the rate of growth in an economy. Then please print it on the accompanying answer sheet. a) an abundance of mineral or petroleum resources b) virtually the same percentage of households in absolute poverty
Encribd is NOT affiliated with the author of any documents mentioned in this site. All sponsored products, company names, brand names, trademarks and logos found on this document are the property of its respective owners.
Question: What should be done with the answer? Answer: Print it on the accompanying answer sheet. | 677.169 | 1 |
surface normal
"Normal vector" redirects here. For a normalized vector, or vector of length one, see unit vector.
A polygon and two of its normal vectors.
A normal to a surface at a point is the same as a normal to the tangent plane to that surface at that point.
A surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a pointP on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.
Calculating a surface normal
For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.
For a plane given by the equation , the vector is a normal. For a plane given by the equation r = a + αb + βc, where a is a vector to get onto the plane and b and c are non-parallel vectors lying on the plane, the normal to the plane defined is given by b × c (the cross product of the vectors lying on the plane).
If a surface S is given implicitly, as the set of points satisfying , then, a normal at a point on the surface is given by the gradient
If a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base. However, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.
Uniqueness of the normal
A vector field of normals to a surface.
A normal to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between the inward-pointing normal and outer-pointing normal, which can help define the normal in a unique way. For an oriented surface, the surface normal is usually determined by the right-hand rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.
Uses
n-dimensional surfaces
The definition of a normal to a two-dimensional surface in three-dimensional space can be extended to -dimensional "surfaces" in -dimensional space. Such a hypersurface may be defined implicitly as the set of points satisfying the equation . If is continuously differentiable, then the surface obtained is a differentiable manifold, and its surface normal is given by the gradient of ,
External link
Question: What is the opposite direction of a surface normal called? Answer: The opposite direction of a surface normal is also a surface normal. | 677.169 | 1 |
What Is the Sum of the Internal Angles in a Hexagon?
Answer
The sum of the internal angles of any hexagon is 720 degrees. This is a polygon characterised with six edges and six vertices. A regular hexagon is made up of sides that have the same length and a cyclic hexagon is any hexagon that is inside a circle.
1 Additional Answer
The sum of the Internal Angles in a hexagon is 720 °. A hexagon has a total of six sides and is indicated mathematically as; a hexagon n = 6. Therefore hexagon has eight right angles but a square has 4 sides, with its interior angles adding up to 360 °.
Question: What is the measure of one internal angle of a regular hexagon? Answer: 120 degrees (720° / 6) | 677.169 | 1 |
Keep in mind I know NOTHING about Vector Algebra other then that I need to use it for this!..
Thanks
April 9th 2013, 07:11 PM
chiro
Re: Calculating the Largest Angle and Direction Between two Planes
Hey rewing.
To get the plane equation, you calculate the normal of the plane and then supply a point to the plane where given a normal n and a point p0, the plane equation is n . (p - p0) = 0 where p is any point on the plane and n is a normal vector with unit length.
I still don't fully understand what the worst angle is though. Can you maybe explain it in mathematical terms (like say an inequality)?
Question: What is the 'worst angle' mentioned in the text? Answer: The 'worst angle' is not explicitly defined in the given text, but it might refer to the largest angle between two planes. | 677.169 | 1 |
Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. Ellipses arise from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.
Elements of an ellipse
The ellipse and some of its mathematical properties.
An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical axes. The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.[1]
The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,[2][3] the major and minor semiaxes,[4][5] or major radius and minor radius.[6][7][8][9]
The foci of the ellipse are two special points F1 and F2 on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF1 + PF2 = 2a ). Each of these two points is called a focus of the ellipse.
Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse.
The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.
The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae).
Drawing ellipses
Pins-and-string method
Drawing an ellipse with two pins, a loop, and a pen.
Question: What are the semi-major and semi-minor axes of an ellipse? Answer: Half of the major and minor axes, respectively. | 677.169 | 1 |
Conic section in rectangular coordinates: (origin at any point in general)
…………………………………………………… eqn(6)
Is the equation of general conic section including circle, ellipse, parabola and hyperbola according as
e = 0, e < 1, e =1, and e > 1. Focus is the point (h, k) and directrix is the st. line . The cases would be clear in the appropriate sections.
Let the point V be the vertex at the origin (0, 0) and let the directrix DD' be at a distance a , i.e., at the point G ( - a, 0). If the focus is at F, VF/VG = e, as VG = a, VF must be ae, so that F is at (ae, 0). If P(x, y) be any point on the conic, PF/PD = e, If PD cuts y-axis at R, or, PF/(PR +RD) = e, or, PF/(PR +VG) = e, or, PF/(VQ +VG) = e, or, r/(x + a) = e,
From here we can go to r/(r cos + a ) = e, ……………………………… eqn.(7)
Or, ……………… ……………………………………………...eqn.(8)
so , or …………………………………………………………………………..eqn(9)
Or ……………………………………………………………… eqn(10)
This 'a' must not be confused with semi major axis of the ellipse. Here it is the distance of the directrix from the vertex. Now l = LF = eSL = e(FV + VG) = e(ae +a);so …………………………..eqn(11)
This is an example of how we derive equations of conic section from focus directrix definition from the first principles.
In the above figure and with the labels as before, A parabola is the curve such that any point on this curve, P(x, y) is at the same distance from the focus F as the distance from the directrix, PD. So PF = PD = QE.
Next, if V(0, 0) is a point on the parabola, called its vertex, then , by above definition, VE = VF. Let us call VE = a, so that F is the point (a,0). Now, PF 2 = r 2 = FQ 2 +QP 2 =
(x-a) 2 +y 2 . Again PD 2 = PR 2 + RD 2 =(x+a) 2
Thus , or y 2 = 4ax …………………………… ………..eqn(12)
It the origin is transferred to the point ( - a , 0), or it is taken to the directrix, then x should be replaced by x – a , so that the equation to the parabola becomes,
y 2 = 4a(x – a) …………………………. eqn(13a)
Question: What is the equation of a parabola given in the text? Answer: y² = 4ax
Question: Which conic sections are included in the given equation? Answer: Circle, ellipse, parabola, and hyperbola | 677.169 | 1 |
Let a st.line VU revolve around a fixed st.line VG making a constant angle with it at V and generate a double cone as shown in the figure. Let a plane parallel to VG and perpendicular to the plane of VUW cut the double cone in a curve in two branches LMN and L'M'N' , M and M' being two points on the cone. Let VC = b , be perpendicular to MM'. Set up a rectangular Cartesian coordinate system at C , midpoint of M'M and MM' being the x-axis and YY' being y-axis in the intersecting plane. Take any point P on the curve of intersection and draw a perpendicular PR onto MM' and extend it until it meets the curve at N. Coordinates of the point P are x = CR and y = PR. Take a plane containing PR and perpendicular to the plane VST . this plane intersects the cone in a circle PST having its centre at O and radius
This can be proved from congruence of the two triangles VSO and VOT, having the side VO common, and two equal angles SVO = = OVT and having a right angle each. OVCR can be proved to be a rectangle as three of its angles are right angles, so that OR = VC = b. Now PR is in the intersecting plane and OR is in the plane of the circle PST and the two planes are perpendicular to each other. So OR PR and it follows that OP 2 = OR 2 + PR 2 .
The relationship between x and y shall be the eqn. to the curve of intersection we require, which can be obtained from this eqn.(a) r in terms of x. We immediately observe that r = OS = OV tan and OV = CR = x. Hence we get, x 2 tan 2 = b 2 + y 2 ………………………………...(b)
is the required equation to the curve of intersection. If we denote
the length CM = a, observe that CMV = MVO = , so that tan = b/a.
Now eqn. (b) becomes, x 2 b 2 = a 2 b 2 + a 2 y 2 or,
…………… .( c)
which is standard equation to a hyperbola . Note that 'a' is actually seen to be its semi-major axis and 'b' is equal to its semi-minor axis, though it is not in the plane of the curve. The value of e may be obtained from ,
………………………………… (d)
which is always greater than 1 for any given acute angle . For a different value of e we need a different value of , or in other words, need a different cone altogether. From one cone, we get hyperbolas all of same e value i.e., sec 2 . This is because either we have to choose a different cone or different values of a and b; to get different hyperbolas from the same cone, i.e., the plane of intersection must be different.
Continued from the previous slide
Question: What are the two branches of the curve of intersection named? Answer: The two branches of the curve of intersection are LMN and L'M'N'. | 677.169 | 1 |
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.
One such regular tetrahedron has a volume of ⅓ of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 1/6 of that of the cube, each.
A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.
If two opposite corners of a cube are truncated at the depth of the 3 vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron.
All but the last of the figures shown have the same symmetries as the cube (see octahedral symmetry).
Question: What is the total number of cubes that can be inscribed in a dodecahedron? Answer: 5 | 677.169 | 1 |
math/geometry pls help A coin of radius 1 cm is tossed onto a plane surface that has been tesselated by right triangles whose sides are 8 cm, 15 cm, and 17 cm long. What is the probability that the coin lands within one of the triangles?
Wednesday, April 21, 2010 at 8:23am by Eunice
math/geometry pls help well, if it is entirely within, it cannot touch any corner or side. STart with the area along the sides the center of the coin cannot be in. The center has to be greater than .5cm from any side. Draw the original triangle. Now mark a line parallel to each side, inside the ...
Wednesday, April 21, 2010 at 8:23am by bobpursley
math geometry help02pm by marko
GEOMETRY PLS HELP46pm by marko
GEOMETRY PLS HELP done
Tuesday, January 31, 2012 at 8:46pm by Reiny
geometry PLS HELP I could not draw the full picture of diagram below i need to explain the relationship between the measure of <abd and measures of <bdc and <dcb d <__a_________b_________c <b measure 59 degree <d measures 62 degree how can i send you a photo of diagram?
Monday, April 2, 2012 at 9:39pm by marko
Question: Who is the author of the first message in the text? Answer: Eunice
Question: What is the topic of the conversation between Eunice and bobpursley? Answer: Geometry, specifically the probability of a coin landing within a right triangle on a tesselated plane surface. | 677.169 | 1 |
It's a pretty common task to get the distance between 2 points. Maybe you're wanting to see if two points are close enough to have collided. Maybe you're making a golf game and the closer the ball is to the hole the higher the score. Either way, you need to know the distance.
The Math
The math for this is related to the Pythagorean Theorem. You know, or can find, the differences between two x values and two y values. What you're looking for is the hypotenuse of the right triangle formed by those legs.
Here's the formula. We'll talk in a bit.
This should look a lot like the formula Pythagorean Theorem, because it is. Distance is C, the difference between x1 and x2 is A, and the difference between y1 and y2 is B.
Everywhere I've seen this formula it's always listed as x2 - x1 and y2 - y1. It really doesn't matter though. Because you're squaring the difference, it will always be a positive number so you don't need to worry about which number is bigger.
In Code
In the examples below we're going to use helper variables just to keep it a bit cleaner.
The Java class also has a few more methods that will let you get the distance from a point to a specific X, Y value without creating a new Point reference and can get the distance squared as well if you need that.
Python
dX = x2 - x1
dY = y2 - y1
distance = math.sqrt(dX * dX + dY * dY)
Other languages?
Pretty much any language that has a function to get the square root of a number will be able to get the distance between two points. Looking at the 3 languages listed above, the formula is the same for all 3. It's just the commands that differ a bit.
You probably learned to measure angles in degrees early in school. A circle has 360 degrees, a perfect corner is a 90 degree angle, and 180 degrees forms a straight line.
1 radian is when the radius is equal to the arc created
But then you took trigonometry or pre-calculus and were introduced to a radian. A radian does the same thing as a degree. It's a way to measure an angle. But instead of going 0 to 360 like degrees, radians go from 0 to .
The image to the right, sourced from Wikipedia, shows that 1 radian is the angle created when the ratio between the radius and arc along the circle is 1.
What happens with most people is that a degree makes more sense than a radian because degree was learned first. Similar idea to going between imperial and metric measurements. If you grew up in the US, measuring in inches probably makes more sense than centimeters. Can't speak for everybody, but I have the same vibe measuring in centimeters and radians compared to inches and degrees.
Why?
Question: What are the differences between the two points used in the formula? Answer: The differences between the two x values (x2 - x1) and the two y values (y2 - y1)
Question: What is the unit of measurement for an angle in radians? Answer: Radians go from 0 to infinity
Question: Which variable represents the distance in the formula? Answer: The variable 'distance' | 677.169 | 1 |
If I want to expand or reduce a shape what mathematical methods are there to do this.
I'd like to understand scaling which seems simple enough. Using my limited knowledge I would do this by measuring the angle and distance of each point from a given anchor point, and then re-plot them by multiplying the distance against a scaling factor.
I have no idea how optimal this method is, or how to express it using mathematical notation, so I'd like to know.
I'd also very much like to understand how to make a shape expand or reduce around it's interior. For example, I thought it would be something like this...
I use a circle of a given radius on each point, and at each point create a bisecting line, then construct the larger or smaller shape using the intersecting points of the circles. However, as you can see this method has numerous errors, shape B obviously has angles that differ from shape A, what's the correct way to expand / shrink a shape in this manner?
Plain english and mathematical notation answers are gratefully requested, I'm still learning a lot of notation.
Update
I'm not sure that the second example is clear enough, so I've made this image to describe what I'm looking for.
Using this example, it's clear that projection scaling isn't going to produce the shape required. what is this sizing method called, and how is it done mathematically?
2 Answers
You can follow your anchor point (projection) approach using a point in the interior. That will preserve angles. It is not clear how you got from A to B in the second drawing. The lines between corresponding vertices do not meet in a point, which is why the angles change.
I've updated the text of my question to provide some clarity for the second part. I'll construct a shape that doesn't scale the way I want using projection if my example isn't one of these.
–
SlomojoJun 28 '11 at 0:12
1
You can preserve angles, but not side ratios, by drawing a parallel to each side offset into the interior by a desired amount, then taking the intersections of the parallels. But if you want similarity (preserve angles and side ratios) your initial approach is the way to go.
–
Ross MillikanJun 28 '11 at 0:17
According to Joeseph O'Rourke's answer, it is the offset curve, related to the Minkowski sum. One formulation is at cagd.cs.byu.edu/~557/text/ch8.pdf. For polygons, you can just find the slope-intercept equation for each side, add or subtract the correct constant from the intercept, then find the intersections to get the new corners.
–
Ross MillikanJun 28 '11 at 12:57
I understand that this is an old question and already has an accepted answer, but for anyone else landing here there is now a great open source library (free for commercial use) that will do polygon offsetting (amongst other things) called Clipper:
Question: How can you mathematically express the scaling method using mathematical notation? Answer: This method can be expressed using the formula: newpoint = anchorpoint + (scalingfactor * (point - anchorpoint))
Question: How can you mathematically describe the offset curve method for polygons? Answer: Find the slope-intercept equation for each side, add or subtract a constant from the intercept, then find the intersections to get the new corners. | 677.169 | 1 |
Counterexample
Throughout Geometry, students write definitions and test conjectures using counterexamples. When writing definitions, counterexamples are useful because they ensure a complete and unique description of a term. If a counterexample does not exist for a conjecture (an if - then statement), then the conjecture is true.
Study Your Way
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Expert teachers who know their stuff bring personality & fun to every video.
Question: What type of statement is a conjecture? Answer: An if-then statement. | 677.169 | 1 |
points, locate a point equidistant from these two points
where this equidistance is rather arbitrary. This new point,
and the given point, both line on a perpendicular so join and
extend them.
There are three main methods of constructing a parallel to a line through
a point (P) not on the line. One is known as the equidistant method,
another the rhombus method, and the third as
two perpendiculars.
The equidistant method uses a perpendicular through the point
not on the line and an additional perpendicular most anywhere (the further
away the better). The distance from point P to the line is transferred
to the other perpendicular and used to construct the parallel.
A rhombus is a quadrilateral with all sides equal. This last fact
means that it is also a parallelogram. Proceed by drawing any
line through P intersecting the line at Q. Now from point Q
draw an arc through P and intersecting the given line at R.
Locate the point S the distance PQ
from P and R by drawing small arcs.
Students have the least difficulty with the two perpendiculars
method so we will keep this discussion short.
Once a perpendicular is constructed through the point not on the
line, then a perpendicular to that perpendicular is constructed
at the given point.
The activity
for this section details the construction of the regular pentagon.
Although the regular heptagon was listed as unconstructable, the
regular heptadecagon (17-gon) is constructible as are any n-gons
where n has only factors of 2m,
primes (possibly more than one, but all different)
of the form 22n+1. The 32 known
constructiblen-gons for odd n can found by using Pascal's
Triangle down to row 32 and treating as binary odd=1 and even=0.
Unless another prime Fermat number
is found, this is all there are.
Now that we have the diagonal equation for finding the number of
diagonals for a given n-gon, we can invert the process,
start with the number of diagonals and find the number of sides,
to practice using the quadratic formula.
Consider an n-gon with 119 diagonals.
Because of my fondness for the heptadecagon, some might quickly guess
n = 17 and be correct.
However, we want a way to solve this kind of problem in general.
First, observe 119 = n(n-3)/2. Vertically under this
distribute the 2: 238 = n2 - 3n and then
n2 - 3n - 238 = 0.
In general there are three
main ways to solve such an equation and find the roots, zeroes,
solutions, or x-intercepts. These three methods are
factoring, graphing, and using the quadratic formula.
Not all quadratics factor and some students spend six weeks in
algebra trying to learn the method. I choose not to formally
Question: Which n-gons are constructible? Answer: Those where n has only factors of 2m and primes of the form 2^2n+1. | 677.169 | 1 |
These four expressions relating the coordinates (x,y) to the coordinates (r,\ue000)apply only when\ue000 is de\ufb01ned, as shown in Figure 3.2a\u2014in other words, when posi-tive\ue000 is an angle measuredcounterclockwise from the positivex axis. (Some scienti\ufb01ccalculators perform conversions between cartesian and polar coordinates based onthese standard conventions.) If the reference axis for the polar angle\ue000 is chosento be one other than the positivex axis or if the sense of increasing\ue000 is chosen dif-ferently, then the expressions relating the two sets of coordinates will change.
Would the honeybee at the beginning of the chapter use cartesian or polar coordinateswhen specifying the location of the \ufb02ower? Why? What is the honeybee using as an origin ofcoordinates?
Quick Quiz 3.1
r\ue000\u221ax2\ue001 y2
tan\ue000\ue000
y
x
y\ue000 rsin\ue000
x\ue000 rcos\ue000
3.1
2.2
Q
P
(\u20133, 4)
(5, 3)
(x, y)
y
x
O
O
(x, y)
y
x
r
\u03b8
(a)
\u03b8(b)
x
r
y
sin\u03b8 =y
r
cos\u03b8=xr
tan\u03b8 =xy
\u03b8\u03b8\u03b8
Figure 3.1Designation of points
in a cartesian coordinate system.
Every point is labeled with coordi-
nates (x,y).
Figure 3.2(a) The plane polar
coordinates of a point are repre-
sented by the distancer and the an-
gle \ue000, where \ue000is measured counter-
clockwise from the positivex axis.
(b) The right triangle used to re-
late (x,y) to (r,\ue000).
You may want to read Talking Apes
and Dancing Bees(1997) by Betsy
Wyckoff.
60
CHAPTER 3Vectors
VECTOR AND SCALAR QUANTITIES
As noted in Chapter 2, some physical quantities are scalar quantities whereas oth-ers are vector quantities. When you want to know the temperature outside so thatyou will know how to dress, the only information you need is a number and theunit \u201cdegrees C\u201d or \u201cdegrees F.\u201d Temperature is therefore an example of ascalar
Question: Which of the following is NOT a valid expression to find y in terms of r and θ? A) y = r sin(θ) B) y = r cos(θ) C) y = r tan(θ) D) y = r^2 sin(θ)
Answer: C) y = r * tan(θ)
Question: What are the coordinates of point P in the given Quick Quiz 3.1?
Answer: The coordinates of point P are (5, 3).
Question: Under what condition do these expressions apply?
Answer: These expressions apply when θ is defined, i.e., when positive θ is an angle measured counterclockwise from the positive x-axis. | 677.169 | 1 |
Related Theorems: Theorem: If A, B, and C are distinct points and AC + CB = AB, then C lies on . Theorem: For any points A, B, and C, AC + CB . Pythagorean Theorem: a2 + b2 = c2, if c is the hypotenuse. Right Angle Congruence Theorem: All right angles are congruent. See proof. Note: While you can usually get away with not knowing the names of theorems, your Geometry teacher will generally require you to know them.
Question: Which theorem is used to find the length of the hypotenuse of a right-angled triangle? Answer: Pythagorean Theorem | 677.169 | 1 |
When two straight lines cross, the opposing angles are equal
An angle drawn in a semi-circle is a right angle
Two triangles with one equal side and two equal angles are congruent
Thales is credited with devising a method for finding the height of a ship at sea, a technique that he used to measure the height of a pyramid, much to the delight of the Egyptians. For this, he had to understand proportion and possibly the rules governing similar triangles, one of the staples of trigonometry and geometry.
It is unclear exactly how Thales decided that the above axioms were irrefutable proofs, but they were incorporated into the body of Greek mathematics and the influence of Thales would influence countless generations of mathematicians.
Pythagoras
Pythagoras (Public Domain)
Probably the most famous name during the development of Greek geometry is Pythagoras, even if only for the famous law concerning right angled triangles. This mathematician lived in a secret society which took on a semi-religious mission. From this, the Pythagoreans developed a number of ideas and began to develop trigonometry. The Pythagoreans added a few new axioms to the store of geometrical knowledge.
The sum of the internal angles of a triangle equals two right angles *(180o).
The sum of the external angles of a triangle equals four right angles (360o).
The sum of the interior angles of any polygon equals 2n-4 right angles, where n is the number of sides.
The sum of the exterior angles of a polygon equals four right angles, however many sides.
The three polygons, the triangle, hexagon, and square completely fill the space around a point on a plane - six triangles, four squares and three hexagons. In other words, you can tile an area with these three shapes, without leaving gaps or having overlaps.
For a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Most of these rules are instantly familiar to most students, as basic principles of geometry and trigonometry. One of his pupils, Hippocrates, took the development of geometry further. He was the first to start using geometrical techniques in other areas of maths, such as solving quadratic equations, and he even began to study the process of integration. He studied the problem of Squaring the Circle (which we now know to be impossible, simply because Pi is an irrational number). He solved the problem of Squaring a Lune and showed that the ratio of the areas of two circles equalled the ratio between the squares of the radii of the circles.
Euclid
Alongside Pythagoras, Euclid is a very famous name in the history of Greek geometry. He gathered the work of all of the earlier mathematicians and created his landmark work, 'The Elements,' surely one of the most published books of all time. In this work, Euclid set out the approach for geometry and pure mathematics generally, proposing that all mathematical statements should be proved through reasoning and that no empirical measurements were needed. This idea of proof still dominates pure mathematics in the modern world.
Archimedes
Question: What is the title of Euclid's landmark work on geometry? Answer: The Elements | 677.169 | 1 |
Trigonometric Values of Angles Study Guide
Trigonometric Values of Angles
Some very interesting and important functions are formed by dividing the length of one side of a right triangle by the length of another side. These functions are called trigonometric because they come from the geometry of a triangle. The domain consists of the measures x of angles. Let H represent the length of the hypotenuse, A represent the length of the side adjacent to the angle x, and the letter O represent the length of the side opposite (away) from the angle x. A right triangle with angle x is depicted in Figure 4.1.
Mnemonic Hint
Some people remember the first three trigonometric functions by saying "Oliver Had A Heap Of Apples" to remember the , , and of sin(x), cos(x), and tan(x). Others say SOA CAH TOA to remember sin(x) = , cos(x) = , and tan(x) = .
The six trigonometric functions, sine (abbreviated sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot), are defined for each angle x by dividing the following sides:
sin(x) =
cos(x) =
tan(x) =
sec(x) =
csc(x) =
cot(x) =
The first thing to notice is that all of the functions can be obtained from just sin(x) and cos(x) using the following trigonometric identities.
Thus, all of the trigonometric functions can be evaluated for an angle x if the sin (x) and cos(x) are known.
The next thing to notice is that the Pythagorean theorem, which, stated in terms of the sides O, A, and H, is O2 + A2 = H2. And, if we divide everything by H2, we get the following:
Thus, (sin(x))2 + (cos(x))2 = 1. To save on parentheses, we often write this as sin2(x) + cos2(x) = 1. Because no particular value of x was used in the calculations, this is true for every value of x.
Drawing triangles and measuring their sides is an impractical and inaccurate method to calculate the values of trigonometric functions. Most people use calculators instead. Although, when using a calculator, it is very important to make sure that it is set to the same format for measuring angles that you are already using: that is, degrees or radians.
There are 360 degrees in a circle, possibly because ancient peoples thought that there were 360 days in a year. As the earth went around the sun, the position of the sun against the background stars moved one degree every day. The 2π radians in a circle correspond to the distance around a circle of radius 1. Because radians already correspond to a distance, there is no need for conversions when calculating with radians. Mathematicians thus use radians almost exclusively.
To convert from degrees to radians, multiply by
To convert from radians to degrees, multiply by
Conversion Hint
Question: How many degrees are there in a circle? Answer: 360 degrees.
Question: What is the Pythagorean theorem stated in terms of the sides O, A, and H? Answer: O² + A² = H².
Question: What are the two common formats for measuring angles? Answer: Degrees and radians. | 677.169 | 1 |
triple product is a product of three vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
Scalar triple product
The scalar triple product (also called the mixed or box product) is defined as the other two.
Geometric interpretation
Geometrically, the scalar triple product
is the (signed) volume of the parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
defined by the three vectors given.
Properties
The scalar triple product can be evaluated numerically using any one of the following equivalent characterizations:
Switching the two vectors in the cross product negates the triple product, i.e.: .
The parentheses may be omitted without causing ambiguity, sincecannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the 3 × 3 matrix having the three vectors as its rows or columns (the determinant of a transposed matrix is the same as the original); this quantity is invariant under coordinate rotation.
Note that if the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the "parallelepiped" defined by them would be flat and have no volume.
There is also this property of triple products:
Scalar or pseudoscalar
Although the scalar triple product gives the volume of the parallelepiped it is the signed volume, the sign depending on the orientation
Orientation (mathematics)
In mathematics, orientation is a...
of the frame or the parity of the permutation of the vectors. This means the product is negated if the orientation is reversed, by for example by a parity transformation, and so is more properly described as aif the orientation can change.
This also relates to the handedness of the cross product; the cross product transforms asunder parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product must be pseudoscalar valued.
, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product
Question: What shape is a parallelepiped? Answer: A three-dimensional figure formed by six parallelograms.
Question: What happens to the scalar triple product if the two vectors in the cross product are switched? Answer: The triple product is negated.
Question: What is another way to evaluate the scalar triple product numerically? Answer: As the determinant of the 3 × 3 matrix having the three vectors as its rows or columns. | 677.169 | 1 |
You, Too, Can Understand Geometry - Just Ask Dr. Math !
Have
Students just like you have been turning to Dr. Math for years asking questions about math problems, and the math doctors at The Math Forum have helped them find the answers with lots of clear explanations and helpful hints. Now, with Dr. Math Introduces Geometry, you'll learn just what it takes to succeed in this subject. You'll find the answers to dozens of real questions from students who needed help understanding the basic concepts of geometry, from lines, rays, and angles to measuring three-dimensional objects and applying geometry in the real world. Pretty soon, everything from recognizing types of quadrilaterals to finding surface area to counting lines of symmetry will make sense. Plus, you'll get plenty of tips for working with tricky problems submitted by other kids who are just as confused as you are.
You won't find a better introduction to the world and language of geometry anywhere! less
Question: What is the tone of the text? Answer: Encouraging and helpful | 677.169 | 1 |
problem (probably two years after I first saw it) I stumbled onto the
solution provided above.
I hope this helps. Please write back if you have any further
questions about this.
- Doctor Greenie, The Math Forum
Date: 12/10/2002 at 19:52:43
From: Javier Vacio
Subject: Mensa: Triangle Area Problem
Hi,
Is there a geometric solution to this problem?
Thank you!
Date: 12/16/2002 at 06:01:07
From: Doctor Floor
Subject: Re: Equilateral Triangle Area Problem
Hi, Javier,
Let me try to offer you a solution. Instead of the lengths 3, 4, and
5, I used the distances d1, d2, and d3 from C, B, and A, respectively,
to a fourth point E in the interior of the equilateral triangle ABC.
First, we rotate the figure of ABC and E about C through 60 degrees,
resulting in an equilateral triangle BHC with G congruent to ABC with
E. In particular we see that CE is rotated through 60 degrees to CG,
which shows that CEG is an equilateral triangle, so that EG = CE = CG
= d1. This triangle I will now refer to as EQUI(d1): an equilateral
triangle with side d1:
Now we draw some simple conclusions:
* The point E divides the original triangle into three triangles,
which I have made into a red one AEC, a yellow one AEB, and a white
one BEC.
* The area of the white and red triangles together is the same as
the area of BEG and CEG together.
* BEG is the triangle with sides equal to the given d1, d2, d3,
which I will from now refer to such a triangle as T.
* We conclude that the white and red triangles together have an area
equal to [T] + [EQUI(d1)] (where [x] is the area of x).
With similar reasoning we can draw similar conclusions about the
areas of the white and yellow triangle together, and about the areas
of the red and yellow triangle together.
Adding the areas of white+red, white+yellow, and red+yellow gives
2*[ABC]. It also gives 3*[T]+[EQUI(d1)]+[EQUI(d2)]+[EQUI(d3)].
So, if we read in areas, we have:
[ABC] = 1.5*[T]+ ([EQUI(d1)]+[EQUI(d2)]+[EQUI(d3)])/2.
Is this enough for a purely geometrical solution of your problem?
If you have more questions, just write back.
Best regards,
- Doctor Floor, The Math Forum
Question: When did Doctor Floor respond to Javier's email? Answer: 12/16/2002
Question: Which two triangles together have an area equal to [T] + [EQUI(d1)]? Answer: The white and red triangles | 677.169 | 1 |
where n > 0 and a and b are the radii of the oval shape. The case n = 2 yields an ordinary ellipse; increasing n beyond 2 yields the hyperellipses, which increasingly resemble rectangles; decreasing n below 2 yields hypoellipses which develop pointy corners in the x and y directions and increasingly resemble crosses. The case n = 1 yields a rhombus. For a = b it is also the unit circle in R2 when distance is defined by the n-norm.
Effects of n
When n is a nonzero rational number p⁄q (in lowest terms), then the superellipse is a plane algebraic curve. For positive n the order is pq; for negative n the order is 2pq. In particular, when a and b are both one and n is an even integer, then it is a Fermat curve of degree n. In that case it is nonsingular, but in general it will be singular. If the numerator is not even, then the curve is pasted together from portions of the same algebraic curve in different orientations.
For example, if x4/3 + y4/3=1, then the curve is an algebraic curve of degree twelve and genus three, given by the implicit equation
or by the parametric equations
The area inside the ellipse can be expressed in terms of the gamma function, Γ(x), as
Generalizations
Example of the generalized superellipse with m ≠ n.
The superellipse is further generalized as:
Superellipsoid
In three-dimensions, a superellipsoid or superegg can be made by revolving a superellipse into a surface of revolution and scaling. Following , it is convenient to distinguish a north-south parameter n, an east-west parameter e, and length, width, and depth parameters ax, ay, az. Then an implicit equation for the surface is
Parametric equations in terms of surface parameters u and v (longitude and latitude) are
Hermann Zapf's typeface Melior, published in 1952, uses superellipses for letters such as o. Many web sites say Zapf actually drew the shapes of Melior by hand without knowing the mathematical concept of the superellipse, and only later did Piet Hein point out to Zapf that his curves were extremely similar to the mathematical construct, but these web sites do not cite any primary source of this account. Thirty years later Donald Knuth built into his Computer Modern type family the ability to choose between true ellipses and superellipses (both approximated by cubic splines).
Question: What is the three-dimensional generalization of a superellipse called? Answer: A superellipsoid or superegg.
Question: What happens to the shape as 'n' increases beyond 2? Answer: The shape increasingly resembles a rectangle, forming hyperellipses. | 677.169 | 1 |
Equilateral Triangles Puzzle
The triangle ABC is an equilateral triangle with an area S and a side length a. The line CF is a continuation of the line AC, AD is a continuation of BA and BE is a continuation of CB. The length of all continuation segments (CF, AD and BE) is a—the same as the length of triangle ABC's sides. The puzzle is to calculate the area of the triangle DEF.
Discuss this Article 2
The Area of an equilateral traingle is equal to the side squared times the square root of 3 quantity over 4. Triangle ABC is equilateral by definition so I will save proving that it is.
Each of the sides are equal to aSIN90 + 2aSIN90, so the area would be calculated in T-SQL something like this:
DECLARE @a DECIMAL
SELECT (POWER((@a*2*SIN(90) + @a*SIN(90)), 2)*SQRT(3))/4 ;
The reason the sides are equal to aSIN90 + 2aSIN90 is that perpindicular line from any of the sides of the external equilateral triangle to the vertices of the internal equilateral will bisect the 120 degree angle there making two right triangles from the scalene triangle having an unknown quanitity for the base and the remaining legs 2a and a respectively. The two triangles would have bases (opposite sides) equal to hypotenuse (2a and a respectively) times SIN 90 as the SIN of an angle is equal to the opposite over the hypotenuse.
So there you have it. DE, EF, and FD are equal to 2aSIN90 + aSIN90 because they are composed of the opposite sides of right triangles having hypotenuses of 2a and a (a + continuation segment, and continuation segments respectively). As SINtheta is equal to opposite over hypotenuse, opposite is equal to hypotenuse times SINtheta, theta in these cases being 90.
So there you have it again
Question: What is the area of triangle DEF? Answer: The area of triangle DEF is not explicitly calculated in the given text.
Question: What is the value of sin(90)? Answer: 1 | 677.169 | 1 |
geometry I am studying for my GRE exam into grad school and it's been a very long time since I've done geometry. I have a problem to which I need to solve for the area of a triangle but I do not have the base. I do have all angles but I cannot remember how to convert angles into their ...
Monday, July 10, 2006 at 7:29pm by Kaytee
geometry, thanks I just wanted to thank everyone that helped me on my geometry problem. I actually found the answer using the pythagorean theorem and knowing ratio rules that apply to triangles. Thanks again.
Thursday, July 13, 2006 at 4:10pm by Kaytee
Geometry help please! What is the length of a diagonal of a square with sides 16 ft long? Round to the nearest tenth. & The length of the hypotenuse of an usosceles right traingle is 30 meters. Find the area of the triangle. Round to the nearest tenth, if necessary. Any help? The diagonal forms two...
Monday, August 21, 2006 at 10:15pm by Maddy!
geometry:linear measure and precision how to find examples of rulesfor this theme I'm not quite sure what information you are seeking. I need more details. However, I searched Google under the key words "linear measure rules geometry" to get these possible sources: http...
Wednesday, August 23, 2006 at 8:02pm by Alma
geometry Dorothy,Multiply 6x9x15 = 4,860.00 I ...
Thursday, December 8, 2011 at 1:39pm by Ms.Fidelusgeometry Geometry riddl How did they like the story about towel manufacturing? Also, where can I find geometry worksheets on plane and simple? ? ? ?
Tuesday, September 26, 2006 at 9:24pm by Debra
Geometry Help with Geometry please! Do you have any specific questions?
Monday, October 16, 2006 at 3:43pm by Margie
Geometry How do you find the area and circumference of a circle. This site has a great explanation for finding area and circumference of a circle.
Thursday, February 1, 2007 at 6:18pm by Kris
Geometry In a regular polygon (equal sides and angles), you use (n-2)180 to find out what the interior angle sum is (n=number of sides in polygon) and you divide by that number by whatever n is to find out the measure of an individual angle, right? Well, there is going to be a problem ...
Thursday, September 27, 2007 at 9:04pm by Emily
Question: What is the measure of an individual angle in a regular polygon with n sides? Answer: The measure of an individual angle in a regular polygon can be found by dividing the sum of the interior angles by the number of sides, n.
Question: What is the formula to find the area of a circle? Answer: The formula to find the area of a circle is A = π * r^2, where r is the radius of the circle.
Question: What is the area of a triangle with all angles known but no base, given in the first post? Answer: The area of a triangle can be found using the formula 1/2 base height. Since the base is not given, we cannot calculate the area with the information provided. | 677.169 | 1 |
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Postulates Teacher Resources
Find Postulates educational ideas and activities
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Help your middle and high schoolers explore geometric terminology. In this geometry lesson, learners use the entire year to create an ABC Geometry book. Students use a list of terms learned in class to create a page for each letter. Students use a rubric to guide their process.
How can we identify shapes in the Universe? High schoolers will compare and contrast elliptic and hyperbolic geometry. They will also explore one possible way to measure the curvature of the Universe, namely, by measuring the sum of the angles in a triangle. Resource links are included.
Tenth graders investigate the early history of geometry. In this geometry lesson, 10th graders investigate translations, rotations, and reflections. They also solve problems with line of symmetry and rotational symmetry while reviewing important symmetry theoremIn this math learning exercise, high schoolers consider Euclid's postulates , which form some of the basic tenants of geometry. This thorough learning exercise has eight problems for the students to solve and includes an answer sheet.
In this geometry worksheet, students solve for missing midpoints, segments and angles using algebraic equations. They use geometric properties to solve these equations. There are 32 questions on 7 different worksheets.
Students investigate spherical geometry using a globe and an apple. In this spherical geometry lesson plan, students translate Euclidean geometry terms to spherical geometry terms using a globe. They answer 3 questions about spherical geometry and use an apple to represent the globe and they cut spherical 'triangles', they score the 'equator' and they cut circles through the 'poles' of the apple better understand the concepts of spherical geometry.
Tenth graders solve problems using the correct tools. In this geometry lesson plan, 10th graders solve problems by constructions and graphing. They use a protractor, compass and straight edge to help create angles and bisectors. The identify the median, altitude and angle bisector of a triangle.
Learners identify the point of intersection of two lines. In this geometry lesson, students identify the pair of coordinates on a coordinate plane where two lines cross. They identify the lines as parallel or perpendicular.
High schoolers explore the concept of congruent triangles. In this congruent triangles instructional activity, students use Cabri Jr. to construct two congruent triangles using circles. High schoolers determine if the two triangles are congruent by using the SSS, SAS, and ASA postulates.
Learners create posters. In this concepts lesson, groups of students use the internet to research different proofs of the Pythagorean Theorem. In addition, they research Greek Mathmaticians. Learners create posters and bulletin boards containing their information.
Question: What is the main focus of the fifth resource? Answer: Investigating spherical geometry using a globe and an apple
Question: Which website is mentioned in the text? Answer: Lesson Planet
Question: What is the main benefit of Lesson Planet for a first-year teacher? Answer: It is a time saver and helps gather ideas from veteran teachers. | 677.169 | 1 |
Triangle Inequality
TopIn a triangle there are three sides, than any one side of a triangle is always smaller than the sum of all the two other sides is the inequalities of triangle. In just opposite case, if any one side of a triangle is always larger than the sum of all the two sides than the triangle cannot be obtained.
In the above given figure, we see that the length of two sides P and Q when added then it is less than the length of O because it violets the definition of triangle inequalities, inequalities of Triangles theorem. We know that the triangle has equal sides, and the angle which is opposite to these sides are also equal. In a triangle if two sides are not equal than the sum of angles opposite to these sides are also not equal. And the angle which is larger is just opposite the larger side.
Other theorem that if two angles in a triangle is not equal then the sum of all the sides opposite these angles are also not equal. Then the larger side is just opposite to the larger angles.
A packed figure consisting of three line segment which is linked end- to- end is known as triangles. Triangles have vertex, base, altitude, median, area, perimeter, interior angles, and exterior angles.
Three vertices are present in a triangle. Out of three sides one side can be taken as base of a triangle. Mostly we will use reference side because it is used in calculating the area of a triangle. And the perpendicular from the base to the opposite side vertex is Altitude of a triangle. And a line from the vertex of a triangle to the mid Point of the opposite sides is a Median of a triangle. There are seven types of triangle which are isosceles, equilateral, scalene, right triangle, obtuse, acute and equiangular.
Question: Which theorem states that if two angles in a triangle are not equal, then the sum of the sides opposite these angles are also not equal? Answer: The theorem about the relationship between angles and sides in a triangle | 677.169 | 1 |
Parts of Circles
When we study Geometry, and look at the Circle, we say that the circle is a round figure, which has its boundary at equal distance from a Point called Centre of the circle. If we look at different parts of the circle we use the terms center, radius, arc, diameter, sector and a Chord. We first talk about the radius of a circle.
If we draw a line from center of the circle to the boundary of the circle,(which we call the circumference of the circle), then the length of this line is called the radius of the circle. The radius remains of fixed length, from where ever we draw it. Sector, arc, radii, diameter are parts of circle.
If we join the two radius, which moves in the opposite direction from the center of the circle is called the Diameter of the circle, which is a part of a circle. We observe that two radius join to form a diameter, so we say that the diameter = radius * 2.
Now we talk about another term related to the circle i.e. chord. We say that the chord is the Line Segment, which starts at one point on the circumference of the circle and ends at another end. There can be infinite number of chords that can be drawn inside the circle. Moreover, we observe that the diameter is also the chord, as it ends at the circumference of the circle.
We also find that the diameter is the longest chord of the circle. If any line is drawn outside the circle, which touches the circle at only one point, then it is called the Tangent drawn on the circle. Any line drawn from the centre of the circle to the Tangent is always perpendicular. So, we say that the radius is perpendicular to the circle.
Centre
Every geometric shape like Circle, triangle, rectangle, square etc has one center and when we rotate any geometric shape around its center, its orientation remains same means length of any straight line from center to its edge remains same, like we have a circle, whose radius is equal to 7 inch, then from center Point to each point of circle, the length of Straight Line is equals to 7 inch because it behave like a Radius of Circle and this center is called as a centre of a circle. For finding the centre of circle, we use following methods.
Method 1: If two edge points of Diameter is given like we have a circle, whose two diameter edge points are (p, q) and (h, k), then Center of Circle C (x, y) is equals to,
x = (p + h) / 2 and y = (q + k) / 2,
These two 'x' and 'y' points are called as a center of circle in this situation.
Suppose we have a diameter, whose edge points are A (4, 7) and B (6, 9), then center of circle–
x = (4 + 6) / 2,
= 10 / 2,
= 5.
and
y = (7 + 9) / 2,
= 16 / 2,
= 8.
Question: What is the relationship between the radius and the tangent of a circle? Answer: The radius is perpendicular to the tangent | 677.169 | 1 |
So, center point of circle x = 5 and y = 8 means C (5, 8) is a center of circle.
Method 2: If a circle equation is given like–
(x – a)2 + (y – b)2 = r2,
Then here value of 'a' and 'b' is behaving like a center of circle and value of center of circle is equals to C (a, b).
Suppose, we have a circle equation–
(x – 5)2 + (y – 7)2 = r2,
Then center point of circle in this situation are x = 5 and y = 7 or C (5, 7) is the center of circle.
Chord
In Geometry, as we know that we study about different shapes. Starting with the basic elements, viz., point, line & plane, some shapes might be rectilinear while some others curved; some might be plane figures & some others are solids. In plane figures some might be 2 dimensional while some others might be 3 dimensional. In spite of all such classifications, we start geometry with a Point which has neither length nor breadth, nor height whereas the next proceeding element, i.e., line is a 1 dimensional concept which has only length.
As we just mentioned that some of the shapes might be rectilinear while others curved, in curved figures we will learn about Circle & its chord. A circle is a closed curved figure which is drawn by joining a number of points which are equidistant from a fixed point. This fixed point is called the Centre of the circle & is denoted by 'O'. The distance from the centre to any point on the circle is called the radius, 'r' of the circle.
Now let us know about the chord of a circle. As stated just above that a circle is formed by joining a number of points, these points when joined form a closed curve which is the boundary of the circle. Now if we join any one point on this curve to the centre of the circle, it is termed as the radius of the circle. But if we join any two points on the curve or the boundary of the circle, it is called the chord of the circle. In simpler words, a chord is a line segment joining any two points on a circle, example: if we take two points A & B on the circle & join them, we say that AB is the chord of the circle. This is all about chords of a circle.
Sector of a Circle
We are already familiar with Circles & some terms associated with them. Let us recall that a Circle is a closed curve formed by joining all the points which are equidistant from a fixed Point, 'O' in its Centre. This fixed point is the centre of the circle.
Question: What is the fixed point from which all points on a circle are equidistant? Answer: Center, 'O'
Question: What is the closed curve formed by joining all points equidistant from the center of a circle called? Answer: Circle
Question: What are the two points on the circle joined by a chord called? Answer: A and B | 677.169 | 1 |
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Planes, Points, Lines, & Angles
Information About This Unit
Description of Learning Unit:
Explore planes, points, lines, and angles in this unit. Hands-on activities will include making use of strings, a collage, magazines, and Twizzlers to work with and identify planes, points, lines, and angles. Worksheets and a video are also included to reinforce the concepts. This unit is designed to last one week, depending on how often you teach math.
Question: What is the primary focus of the unit? Answer: Explore planes, points, lines, and angles | 677.169 | 1 |
what is a 90 degree angle triangle called?
Possible Answer
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. As a rotation, a right angle corresponds to ... - read more
Any triangle with a 90 degree angle is called a right triangle. Triangle with an angle equal to 90 degrees? A right angle triangle. Does right angled triangle has 90 degrees? All triangles have 180o. A right-angled triangle has one right angle
Question: Is a right triangle also known as a right-angled triangle? Answer: Yes | 677.169 | 1 |
A real line is - in fact, some of you in fancy restaurants might see these kind of breadsticks. This is still not capturing the spirit of the line because the line actually has no thickness. So even as delicious as these are - and these are really good and crunchy. Can you hear the crunch on that? Umm, it is delicious, but they capture the spirit of "lineness," until we draw them like this of course. But understand that actually a line has no thickness. A line just has basically a direction, which is straight. So, I'll put these culinary treats away for a second and clean up my dining room table here. Just come into my home, make yourself at home. Really, lines are objects that look like this. And you'll notice they have - well, an interesting form, because they're very, very straight, and yet there's something to them. There's about two things that I think are interesting about lines. See - what identifies that line out of all the universe of lines? How would we actually find that particular line?
Well, the answer is that I actually need two, two pieces of information in order to completely identify this line. And I'm going to tell you what those two pieces of information are right now, but you can probably figure it out. One is going to be pitch. For example, suppose I take a look at an axis here and I draw a line. Look at this. Ping. That is a beautiful line. But you see that line - I can make it look a little different by doing that. That's now a different line. The pitch is different. In fact, if you were climbing up this, this is going to be a mountain that's not so hard to climb but what about this one. Wow! Would you want to climb up that? I should say not. At least, I wouldn't because it would be really exhausting. This is very heavily pitched; whereas, this is less pitched. So one variable in figuring out a line is to know its pitch, which we call slope.
Question: What is the main characteristic of a real line? Answer: A real line has no thickness. | 677.169 | 1 |
There is a question using both the law of Sines and Cosines, and first you solve for a side and use it to find an angle. Say the side is an irrational number (ex. 10.254873209). Can a student round it to say the nearest hundredth and use it to solve an angle OR should a student use the 10.254873209 to solve the angle?
I have always taught students to use the number and not to round it until the end (the answer) but just wondered if that is allowed.
Question: What is the benefit of waiting to round the answer until the end of the calculation? Answer: Waiting to round the answer until the end of the calculation ensures that any rounding error is minimized and the final answer is as accurate as possible. | 677.169 | 1 |
Maths - Geometry
Geometry is concerned with the properties of space and the shapes and relationship of things in it. An important topic for this site. Its interesting how much of maths is related to geometry. If an algebra can be any set of objects represented by abstract symbols and a set of rules, the only criteria is that the algebra is consistent (has no contradictions) no requirement to represent reality, so why does so much of it have a geometrical interpretation?
I like to think that at its simplest level:
Algebra - evolved from counting
Geometry - evolved from measuring
Counting may seem more abstract and fundamental than measuring but many of its most fundamental concepts:
Irrational Numbers
√2 length of hypotenuse of triangle whose other sides have len=1
π ratio of circumference to diameter of circle.
Vectors
all seem to be studied first in geometry.
Traditional Geometry
The traditional geometry that many of us were taught at school involves two dimensional constructions with points, lines between points, angles between lines and trigonometry. This type of geometry is very useful and we use a lot of it on this site (here), examples are, proving Pythagoras theorem and the double angle formula.
Euclid was a Greek Mathematician from about 300 BC who unified what was then known about geometry and derived results from 5 postulates:
A straight line may be drawn from any one point to any other point (any 2 points determine a unique line).
A finite straight line may be produced to any length in a straight line.
A circle may be described with any centre at any distance from that centre.
All right angles are equal.
If a straight line meets two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.
The Thirteen Books of Euclid's Elements: Book 1.
We now know about other types of geometry (known as non-Eucliden geometries) which can be derived by changing these postulates. If we change the 5th postulate, known as the parallel postulate, we get hyperbolic geometry.
Vector Space
There are disadvantages to the above approach; we need to work out a diagram or construction for each problem and it gets a lot more complicated if we want to work in three or more dimensions.
Vectors allow us to treat geometry problems in a more analytical way, allowing us to translate to an algebraic approach. However the disadvantage with vectors is that we need to choose a coordinate system which can be quite arbitrary and the results that we are looking for can get hidden by arrays of numbers which depend on the coordinate system that we chose.
There are ways to make our results independent of the coordinate system by expressing results in terms of general basis vectors by using tensor or geometric algebra.
Question: What is the difference between Euclidean and non-Euclidean geometries? Answer: Non-Euclidean geometries can be derived by changing the postulates of Euclidean geometry, particularly the parallel postulate.
Question: Which mathematician unified the known concepts of geometry around 300 BC? Answer: Euclid
Question: Which famous theorem is proven using traditional Euclidean geometry on this site? Answer: Pythagoras theorem | 677.169 | 1 |
Problem: The Best Angle of View
You are standing on level ground in front of a billboard. When
you look up at it, the top of the billboard measures a
feet up the support from eye level and the bottom of the billboard
measures b feet up the support from eye level. You wish
to position yourself in order to maximize your "viewing angle"
(the angle between the lines of sight of the top and the bottom
of the billboard). However, a storm has tilted the billboard,
as shown in the accompanying figure. Find the distance x that
maximizes ß.
Question: What is the effect of the storm on the billboard? Answer: The storm has tilted the billboard. | 677.169 | 1 |
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