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Australian Education - ubd high school geometry congruence
6 Tagged Resources - "ubd high school geometry congruence"
maths homework helper algebra help math software from teachers choice software - math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math problems instantly... How to solve a triangle given two sides and a internet maths tutor. HOTmaths is a set of interactive maths tutorials modelling the methods used by teachers. While it doesn't replace a teacher or tutor, it supplements your learning experience at school, and helps you correct problems with your maths... How to do Decimals, Help with Long
Question: Does the text provide information on how to solve a triangle given two sides and an angle? Answer: Yes, "How to solve a triangle given two sides and an internet maths tutor" | 677.169 | 1 |
that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...
; that is, that it cannot be constructed with ruler and compass.
Solutions
Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve the cissoid of Diocles
Cissoid of Diocles
In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the...
A conchoid is a curve derived from a fixed point O, another curve, and a length d. For every line through O that intersects the given curve at A the two points on the line which are d from A are on the conchoid. The conchoid is, therefore, the cissoid of a circle with center O and the given curve...
In geometry, the Philo line is a line segment defined from an angle and a point. The Philo line for a point P that lies inside an angle with edges d and e is the shortest line segment that passes through P and has its endpoints on d and esolved the problem in the fourth century B.C. using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
False claims of doubling the cube with compass and straightedge abound in mathematical crank
Crank (person)
"Crank" is a pejorative term used for a person who unshakably holds a belief that most of his or her contemporaries consider to be false. A "cranky" belief is so wildly at variance with commonly accepted belief as to be ludicrous...
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
Question: Who solved the problem of doubling the cube in the fourth century B.C.? Answer: The solution was determined by a certain point as the intersection of three surfaces of revolution.
Question: What is the cissoid of Diocles used for? Answer: It can be used to double a cube.
Question: What is the Philo line defined from? Answer: An angle and a point. | 677.169 | 1 |
Newsflash!
The Mathematics behind the Math Midway
October 30th, Lawrence Hall of Science
Monthly Problem
Four Leaf Clover
Around the outside of a 4 by 4 square, construct four semicircles (as shown in the figure) with the four sides of the square as
their diameters. Another square, ABCD, has its sides parallel
to the corresponding sides of the original square, and each side
of ABCD is tangent to one of the semicircles. What is the area
of the square ABCD?
- from AJHSME 1994
The Oakland/East Bay Math Circle Program is made possible by the generous contributions of our many Sponsors.
Question: What is the relationship between the sides of square ABCD and the semicircles? Answer: Each side of ABCD is tangent to one of the semicircles | 677.169 | 1 |
Now, we know that this is the ratio between the two so we don't know, you know this could be 10 and this could be 26. This could be 1 and this could be 13/5, who knows, but it actually doesn't matter because that's what's needed by trigonometry. It's all about the ratios. So, let's just assume that this is 5, that the opposite is equal to 5and the hypotenuse I equal to 13. So, if the opposite is 5 and the hypotenuse is 13 what will be the adjacent be equal to? Well, you could use it by tag and theorem. So, we could say the adjacent squared, right a² + the opposite squared so +5² + 25 = 13², 13² is 169 I think. If you subtract 25 from both sides of this equation, you get a² = 144, a = 12. So, a = 12. And we don't know that a is definitely equal to 12 but we know that the ratio of say the opposite to adjacent is 5 to 12 because we just assumed that the opposite is 5.
Anyway so, they will know what are cos of x and tan x. Socatoa, cos of x is equal to the adjacent over the hypotenuse. The adjacent is 12, hypotenuse is 13 = 12/13. That's the cos of x and the tan of x = opposite over adjacent (toa). So, opposite is 5, adjacent is 12. Go to 5/12 and we'll see what choice that is, that's choice a, cos of x is 12/13, tan x = 5/12.
Next question, oh looks like they want us to learn a lot of trigonometry in geometry which is good. This can warm you up for the trig. In the figure below sin a = 2.7 so this is called this angle A. The side of that is equal to 0.7. They say what is the length of AC? So we want to know that. Let's call that x right. So Socatoa, So tells us that sin of sum angle, let's call that data is equal to the opposite over the hypotenuse. So, sin A in this example. Sin A is going to be equal to the opposite 21 over the hypotenuse, over x and it tells that the sign of A is equal to 0.7 so that's also equal to 0.7. So now, we could just solve this equation for x and we're done. Let's see, so if you multiply x times both sides, you get 21 = 0.7x, 21/0.7 = x. So x = 30 and that's length AC. That's choice C.
Question: In the second scenario, what is the given length of the side opposite to angle A? Answer: 21
Question: What are the given lengths of the opposite and hypotenuse sides in the first scenario? Answer: 5 (opposite) and 13 (hypotenuse)
Question: What is the length of side AC in the second scenario? Answer: 30 | 677.169 | 1 |
Ancient builders and surveyors needed to be able to construct right angles in the field on demand. The method employed by the Egyptians earned them the name "rope pullers" in Greece, apparently because they employed a rope for laying out their construction guidelines. One way that they could have employed a rope to construct right triangles was to mark a looped rope with knots so that, when held at the knots and pulled tight, the rope must form a right triangle. The simplest way to perform the trick is to take a rope that is 12 units long, make a knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop, as shown in the animation. However, the Egyptian scribes have not left us instructions about these procedures, much less any hint that they knew how to generalize them to obtain the Pythagorean theorem: the square on the line opposite the right angle equals the sum of the squares on the other two sides. Similarly, the Vedic scriptures of ancient India contain sections called sulvasutras, or "rules of the rope," for the exact positioning of sacrificial altars. The required right angles were made by ropes marked to give the triads (3, 4, 5) and (5, 12, 13).
In Babylonian clay tablets (c. 1700–1500 bce) modern historians have discovered problems whose solutions indicate that the Pythagorean theorem and some special triads were known more than a thousand years before Euclid. A right triangle made at random, however, is very unlikely to have all its sides measurable by the same unit—that is, every side a whole-number multiple of some common unit of measurement. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of incommensurability.
Locating the inaccessible
By ancient tradition, Thales of Miletus, who lived before Pythagoras in the 6th century bce, invented a way to measure inaccessible heights, such as the Egyptian pyramids. Although none of his writings survives, Thales may well have known about a Babylonian observation that for similar triangles (triangles having the same shape but not necessarily the same size) the length of each corresponding side is increased (or decreased) by the same multiple. A determination of the height of a tower using similar triangles is demonstrated in the figure. The ancient Chinese arrived at measures of inaccessible heights and distances by another route, using "complementary" rectangles, as seen in the next figure, which can be shown to give results equivalent to those of the Greek method involving triangles.
Estimating the wealth
Question: What method did ancient Egyptian builders use to construct right angles in the field? Answer: They employed a rope with specific knots to form a right triangle.
Question: When was the Pythagorean theorem discovered? Answer: It was known to the Babylonians around 1700-1500 BCE, more than a thousand years before Euclid.
Question: What is the relationship between corresponding sides of similar triangles? Answer: The length of each corresponding side is increased (or decreased) by the same multiple. | 677.169 | 1 |
Unit Q Concept 5 - Trigonometric Identities
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Unit Q Concept 5 - Trigonometric Identities
using various simplification methods to verify and simplify trigonometric expressions and equations.
1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.
2. Look for opportunities to:
? Look for a GCF
? Substitute an identity
? Multiplying by conjugate
? Combining fractions w/ binomial denominator
? Separating fractions with monomial denominators
? Factoring
3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines
pair up well, as do secants and tangents, and cosecants and cotangents
4. If the preceding guidelines do not help, try converting all terms to sines and cosines.
5. Always try SOMETHING. Even making an attempt that leads to a dead end provides insight.
Question: Which trigonometric functions pair up well? Answer: Sines and cosines, secants and tangents, cosecants and cotangents | 677.169 | 1 |
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others solids and Kepler-Poinsot polyhedra — are arranged into dual pairs.
Duality is defined in terms of polar reciprocation about a given sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere
x2 + y2 + z2 = r2,
the vertex
(x0, y0, z0)
is associated with the plane
x0x + y0y + z0z = r2.
The vertices of the dual, then, are the reciprocals of the face planes of the original, and the faces of the dual lie in the reciprocals of the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual. This can be generalized to n-dimensional space, so we can talk about dual polytopes. Then the can be defined similarly.
Notice that the dual of a polyhedron will depend on what sphere we reciprocate with respect to, the resulting forms being distortions of one another. The center of the sphere It can be shown that all polyhedra can be distorted into a canonical form where a midsphere exists and in fact the points where the edges touch it average out to give the center of the circle, and this form is unique up to congruencies.
If a polyhedron has an element passing through the center of the sphere, it will have an infinite dual. It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere, and the corresponding graph formed by the dual of this polyhedron is its dual graph.
The concept of duality here is also related to the duality in projective geometry, where lines and edges are interchanged; and is in fact a particular version of
Question: How is the dual of a polyhedron defined in terms of polar reciprocation? Answer: Each vertex of the original polyhedron is associated with a face plane of the dual, and the ray from the center to the vertex is perpendicular to the plane.
Question: What is the equation of the sphere used for polar reciprocation? Answer: x² + y² + z² = r² | 677.169 | 1 |
Welcome to Geometry. If you need to email just click Email to the left. If you would like to get updates when I update this page, just click on Subscribe for updates to the left.
Geometry
Geometry focuses upon relations, properties, and measurement of surfaces, lines, and angles in one, two, and three dimensional figures as investigated and used in this course. It is designed to develop deductive reasoning and to emphasize problem solving using informal proofs and definitions while integrations of algebraic concepts. Successful completion of Algebra 1 is a prerequisite to Geometry.
Question: What is the primary goal of this Geometry course? Answer: To develop deductive reasoning and to emphasize problem-solving using informal proofs and definitions while integrating algebraic concepts. | 677.169 | 1 |
english this is for the book to kill a mockingbird what is significant about Mr. Cunningham kneeling down and saying to Scout "i'll tell him you said hey, little lady"
math given the points A(0,0), B(3,1) and C (1,4) what is the measure of angle ABC I plotted it on a gragh and got 1.46 degress. I am unsure if I am doing this question correctly. Or am I misreading it thanks in advance .
Math base of large triangle is to base of small triangle as 96 miles is to 31 miles. base of large triangle is to base of small triangle as leg (m) of large triangle is to leg (31) of small triangle Does this help?
Math If an area of a parallelogram is 8x+2 with a height of 4. What is the length of the base?
math If the distance between location A to B is straight east 10 km. then from B to C is Nw 18KM How far would you be from point C to A. Would you not need to more information then this to calculate this answer?
math A plane leaves hamilton and flies due east 75 km at the same time a second plane flies in a direction of 40degress southeast for 120 km how far apart are the planes when they reach their destination. I say the calculation is 75^2+120^2-2(75)(120)cos40 the book says to use cos ... t...
math Susan Borrowed $5000. The term of the loan is 12% compouned monthly for 3 years. what is the monthly payments? How much must she pay at the end of of 1 year to pay the balance off? How much did she save in interest by paying the loan off in one year??
Question: Can you calculate the distance from point C to A, given that the distance from A to B is straight east 10 km, and from B to C is northwest 18 km? Answer: No, additional information is needed to calculate this distance, such as the coordinates of points A, B, and C or the exact direction of the northwest path.
Question: What is the distance between the two planes when they reach their destinations, given that one plane flies due east 75 km and the other flies 40 degrees southeast for 120 km? Answer: The distance can be calculated using the formula √(75^2 + 120^2 - 2(75)(120)cos(40)), which gives approximately 109.5 km. | 677.169 | 1 |
Point
posted on: 13 Apr, 2012 | updated on: 20 Dec, 2012
The word Point is defined as the place, position or location in the space. There is no width, length and height of a point, it is also dimensionless. There is no dimension for the indication of point. Dimensionless means they do not have any volume, area, length. If we locate a Point on a simple or standard plane then the point is known as origin.
Suppose the tip of a pencil, corner of a Cube, or a dot on a sheet of paper. In two- dimension, the point is represented by an Ordered Pair. Suppose a point (a, b), where 'a' denotes the horizontal lines and 'b' represents the vertical lines. Many of the construction in the point Geometry consist of many more points. If we are going to define points, then any two points can be connected by a straight line.
Point word was neither complete nor definitive. It only assumed facts about that what the point represents. The notation of points is generally defined in the geometry and topology. Suppose the teacher writing on the blackboard and fills the full board, then it also indicate the point.
The fixed-point number is a number which presents the real thing i.e. data, collection of 5 books that means fix number of books, or we can say collection of fix number after a decimal point. Suppose we have value 2.123 then the point indicates that the 3 number after a decimal number and 1 number before a decimal number. It is also taken for the representation of the floating point number.
We can represent fraction values with the help of points. Point increases the performance or accuracy of a number. The value which we are taken for data types is only an Integer value. If we are having 4 point on a plane surface, and if we join all these four points then we get a line, it means the Combination of points also give us a line. It is necessary to understand that a point is not a thing but it is a place.
We represent the point by placing a dot with a pencil. Dot point has a Diameter, its diameter is around 0.22, but the point doesn't have any size. Mainly upper-case letter were taken to represent a point. If there are many points and these entire point lie in a Straight Line, then they are known as collinear and these points lie on the same plane then it is known as coplanar.
The basic elements of Geometry which form its foundation are points, lines and planes. A Point is a dot made on a plane by a sharp, pointed object, may be the tip of a pen or a pencil on a paper or even a hole made by piercing a pin in a paper. Points are represented by some capital letter. A point has neither length, nor breadth, nor thickness.
Question: Can a point be a thing or is it always a place? Answer: It is always a place, not a thing | 677.169 | 1 |
Geometry: Special Triangles
On this page we hope to clear up problems that you might have
with special triangles, such as a 30°-60°-90°, and
theorems that apply to them, such as the Pythagorean Theorem. Read
on or follow any of the links below to start better understanding special
triangles.
Pythagorean Theorem
One of the most famous mathematicians who has ever lived,
Pythagoras, a Greek scholar who lived way back in the 6th
century B.C. (back when Bob Dole was learning geometry), came
up with one of the most famous theorems ever, the
Pythagorean Theorem. It says — in a right
triangle, the square of the measure of the hypotenuse equals
the sum of the squares of the measures of the two legs. This
theorem is normally represented by the following equation:
a^2 + b^2 = c^2, where c represents the hypotenuse.
With this theorem, if you are given the measure of two sides of a triangle,
you can easily find the measure of the other side.
Example:
1. Problem: Find the value of c in the accompanying figure.
Solution: a^2 + b^2 = c^2 Write the Pythagorean
Theorem and then plug in
any given information.
5^2 + 12^2 = c^2 The information that was
given in the figure was
plugged in.
169 = c^2 Solve for c.
c = 13
45-45-90 Triangles
One of the special right triangles which we deal with in geometry
is an isosceles right triangle. These triangles are also
known as 45-45-90 triangles (so named because of the
measures of their angles). There is one theorem that
applies to these triangles. It is stated below.
In a 45-45-90 triangle, the measure of the hypotenuse is
equal to the measure of a leg multiplied by SQRT(2).
30-60-90 Triangles
There's another kind of special right triangle which we deal
with all the time. These triangles are known as
30-60-90 triangles (so named because of the measures of
their angles). There is one theorem that applies to these
triangles. It is stated below.
In a 30-60-90 triangle, the measure of the hypotenuse is two times
that of the leg opposite the 30° angle. The measure of
the other leg is SQRT(3) times that of the leg opposite the
30° angle.
Trigonometric Ratios
While the word trigonometry strikes fear into the hearts
of many, we made it through (amazing as it may seem to us), and hope
to help you through it, too! Each of the three basic
trigonometric ratios are shown below. There is also an
Question: Who was the mathematician who discovered the Pythagorean Theorem? Answer: Pythagoras | 677.169 | 1 |
Anyway, here's how the theorem works. To keep the numbers simple, let's say a = 3 yards and b = 4 yards. Then to figure out the unknown length c, we don our black hoods and intone that c2 is the sum of 32 plus 42, which is 9 plus 16. (Keep in mind that all of these quantities are now measured in square yards, since we squared the yards as well as the numbers themselves.) Finally, since 9 + 16 = 25, we get c2 = 25 square yards, and then taking square roots of both sides yields c = 5 yards as the length of the hypotenuse.
This way of looking at the Pythagorean theorem makes it seem like a statement about lengths. But traditionally it was viewed as a statement about areas. That becomes clearer when you hear how they used to say it:
"The square on the hypotenuse is the sum of the squares on the other two sides."
Notice the word "on." We're not speaking of the square "of" the hypotenuse — that's a newfangled algebraic concept about multiplying a number (the length of the hypotenuse) by itself. No, we're literally referring here to a square sitting on the hypotenuse, like this:
Let's call this the large square, to distinguish it from the small and medium-sized squares we can build on the other two sides:
Then the theorem says that the large square has the same area as the small and medium squares combined.
For thousands of years, this marvelous fact has been expressed in a diagram, an iconic mnemonic of dancing squares:
Viewing the theorem in terms of areas also makes it a lot more fun to think about. For example, you can test it — and then eat it — by building the squares out of many little crackers. Or you can treat the theorem like a child's puzzle, with pieces of different shapes and sizes. By rearranging these puzzle pieces, we can prove the theorem very simply, as follows.
Let's go back to the tilted square sitting on the hypotenuse.
At an instinctive level, this image should make you feel a bit uncomfortable. The square looks potentially unstable, like it might topple or slide down the ramp. And there's also an unpleasant arbitrariness about which of the four sides of the square gets to touch the triangle.
Guided by these intuitive feelings, let's add three more copies of the triangle around the square to make a more solid and symmetrical picture:
Now recall what we're trying to prove: that the tilted white square in the picture above (which is just our earlier "large square"— it's still sitting right there on the hypotenuse) has the same area as the small and medium squares put together. But where are those other squares? Well, we have to shift some triangles around to find them.
Think of the picture above as literally depicting a puzzle, with four triangular pieces wedged into the corners of a rigid puzzle frame.
Question: Which of the following is NOT a way to test the theorem? (a) Using algebra (b) Building the squares out of crackers (c) Solving a word problem (d) Proving it using calculus Answer: (d) Proving it using calculus
Question: What is the length of the hypotenuse 'c' in the given example? Answer: c = 5 yards | 677.169 | 1 |
Octagon calculator
In geometry, an octagon (from the Greek okto, eight) is a polygon that has eight sides. A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon).
Question: How many lines of reflective symmetry does a regular octagon have? Answer: 8 | 677.169 | 1 |
directrixrix is discussed in the following articles:
cones
...the surface traced by a moving straight line (the generatrix) that always passes through a fixed point (the vertex). The path, to be definite, is directed by some closed plane curve (the
directrix), along which the line always glides. In a right circular cone, the
directrix is a circle, and the cone is a surface of revolution. The axis of this cone is a line through the vertex and...
cylinders
...that is traced by a straight line (the generatrix) that always moves parallel to itself or some fixed line or direction (the axis). The path, to be definite, is directed along a curve (the
directrix), along which the line always glides. In a right circular cylinder, the
directrix is a circle. The axis of this cylinder is a line through the centre of the circle, the line being...
ellipses
...base, the axis, or an element of the cone. It may be defined as the path of a point moving in a plane so that the ratio of its distances from a fixed point (the focus) and a fixed straight line (the
directrix) is a constant less than one. Any such path has this same property with respect to a second fixed point and a second fixed line, and ellipses often are regarded as having two foci and
Question: What is the constant ratio in the definition of an ellipse? Answer: The constant ratio is less than one. | 677.169 | 1 |
I am homeschooling my son and these problems are giving him a difficult time,
I don't know much about math, history is my skill, so if you could help me, I
would most appreciate it.
Directions- complete and draw
1. Given a segment AB, construct and label the locus of points at a distance
AB from point A and equidistant from A and B.
Directions- describe each locus of points completely.
1. In a plane, the locus of the centers of all circles with radius 3cm that
are tangent to a given line L
2. In a plane, the locus of points 4cm from the center of a circle whose
radius is 5cm.
Question: What is the radius of the circle in the third locus of points? Answer: 5cm. | 677.169 | 1 |
They have regular polygonal bases and square sides: from left to right, the bases are triangular, squared, pentagonal, and heptagonal. This series of solids can grow up to infinite sides, and this is the reason these polyhedra have been excluded from the Jonson solids list. Note that the squared base prism simply is... a cube: in fact it has two bases and four faces that all are squares.
The following are two other kind of prisms: the left one is a rectangular cuboid, while the right one is a triangular prism with rectangular (not squared) faces.
The base of a prism can also be a not regular polygon: the model below on the left is an example that has squared faces (a prism can have also different length base edges, with or without one or more squared faces).
The model above on the right is a prism too, even it has two faces are not rectangular. So the prisms family has a lot of different shape solids... but be careful: not all seems to be a prism actually is!
The model above on the left has to squared bases perfectly aligned the one with respect the other, but the edges don't connect the corresponding vertices: the lateral faces are skew polygons, that is they don't lie in a flat plane, but zigzags in three dimensions. The model above on the right doesn't show this problem (the face edges are flat), but the cross-sections parallel to the base faces are not all the same (at mid height the section is a square).
A final definition about prisms: the Parallelepipeds are those with six parallel faces (as the cube or, in general, the rectangular cuboids we saw above). Let's see the following model:
This model of Prism (or more precisely of Parallelepiped) with six rhombic faces is a Rhombohedron: I show it in two versions, where the right one has yellow panels installed to better show the shape of the faces.
Antiprisms are polyhedra composed of two parallel copies of some regular polygon, connected by an alternating band of triangles. Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. Here are two examples, with squared and pentagonal bases:
A particular case is the triangular antiprism, which has a total of eight triangles. Being all these triangles equilateral, the polyhedron actually is an Octahedron (a Platonic solid). I show it with horizontal bases (on the left) and raised up to look similar to what we normally imagine to be an octahedron:
As for the prisms, also antiprisms can grow up to infinite sides (actually both prisms and antiprisms are excluded from the Jonson solids list).
Two models more:
The left one is an antiprism where the triangular faces are not equilateral. The right model has the bases aren't regular polygons, so it can be easily seen the two bases are not identical each other: in fact, this IS NOT an antiprism!
Question: What are the bases of an antiprism? Answer: Two parallel copies of some regular polygon.
Question: Which of the following is NOT a prism? A) Rectangular cuboid B) Triangular prism with rectangular faces C) A solid with squared faces but non-aligned edges D) A solid with two non-rectangular faces Answer: C) A solid with squared faces but non-aligned edges.
Question: What are the bases of the first four solids in the series described? Answer: Triangular, square, pentagonal, and heptagonal. | 677.169 | 1 |
As a conclusion of this article I'd like to collect some information about Platonic Solids.
- Tetrahedron: also an triangular pyramid
- Hexaedron (Cube): also a square prism
- Octahedron: also a square bypyramid, or a triangular antiprism
- Icosahedron: also the sum of a pentagonal bipyramid and a pentagonal antiprism
- Dodecahedron: unfortunately there is not a direct connection with this solid and the polyhedra classes described here.
Question: How many faces does an Octahedron have? Answer: 8 | 677.169 | 1 |
13.2012
ode to equilateral
Yeeaaahh....
I found my favorite triangle.
What?
You don't have a favorite triangle?
Let me introduce you to my friend the Equilateral triangle.
Seriously, you've made things with the others.
Probably Isosceles. Maybe Scalene for you modern improv folks.
You've maybe had some trouble with the points?
You've maybe had to redo a few when things weren't lining up right?
Maybe not?
Maybe it's just me?
Can I tell you that equilateral triangles just line up.
There is no 1/4 inch overlap to worry about!
I know it's math,
but to me, its like magic!
Here's the back. Some cute DS Quilts, some Meadowsweet, and some Groovy Guitars.
Question: How does the speaker describe the process of using equilateral triangles? Answer: Like magic | 677.169 | 1 |
You can put this solution on YOUR website! The sum of the exterior angles in any convex polygon is 360, so you can divide 360 by the number of vertices (sides) to obtain the degree measure of each exterior angle (e.g. for a pentagon, exterior angle = 360/5 = 72). To find the interior angle, simply subtract this angle measure from 180.
Alternatively, you can use the formula 180(n-2)/n to find the interior angle.
Question: If you want to find the interior angle of a polygon, what should you subtract from 180 degrees? Answer: The exterior angle measure | 677.169 | 1 |
plug in the values for x and y to find the perimeter (sum of the sides)
Vectors/286948: Let vector v=<5,-1,4>. Give a vector w in the opposite direction of v and with magnitude twelve. Give its components to the nearest tenth. I don't understand how to solve the problem and I can't seem to find any similar examples in the text. 1 solutions Answer 208093 by scott8148(6628) on 2010-03-30 17:28:14 (Show Source):
w = <-5k, 1k, -4k>
the "k" is the factor to make the magnitude of w equal 12
the magnitude is the length of the vector
___ found by using the Pythagorean theorem on the components
___ m^2 = x^2 + y^2 + z^2
144 = 25k^2 + k^2 + 16k^2 = 42k^2
k = sqrt(144/42) = 1.852 (approx)
plug in the value of k to find the components of w
Triangles/286961: I have a triangle with given angles 135, and 30. The side opposite the 135 degree angle is 12 units. The questions is: Find the length of the side opposite the 30 degree angle, x. 1 solutions Answer 208088 by scott8148(6628) on 2010-03-30 17:10:27 (Show Source):
Rate-of-work-word-problems/286967: Working by herself, Mary requires 16 minutes more than Antoine to solve a mathematics problem. Working together, Mary and Antoine can solve a problem in 6 minutes. If this situation is represented by the equation (6/t)+(6/t+16)=1
, where t represents the number of minutes Antoine works alone to solve the problem, how many minutes will it take Antoine to solve the problem if he works by himself?
You can put this solution on YOUR website! the y-value increases by 2 when the x-value increases by 1 ___ this is called the SLOPE of a linear function
___ the slope is usually represented by the letter "m"
there is also a "fudge factor" added (or subtracted) to make the values match up
___ this number is is the y-intercept of the linear function ___ it is the point on the y-axis where the line crosses
___ the intercept is usually represented by the letter "b"
Question: What is the value of k in vector w? Answer: 1.852 (approximately)
Question: What is the slope of the linear function described in the text? Answer: m | 677.169 | 1 |
from KET illustrates how an origin is used for positive and negative measurement along a straight line and on a flat plane. It also shows how an origin, latitude, and longitude identify locations on Earth and explores how measuring temperature differs from measuring height or weight.
Origin is a mathematical construct used to find an exact location or measurement along a line, on a flat plane, or in three-dimensional space. Identifying an origin, the point from which measurements originate, gives us a common frame of reference to describe a location or measurement.
Temperature scales provide a practical application of the concept of origins. The Fahrenheit and Celsius scales assign arbitrary numbers to the temperatures of certain natural phenomena such as the freezing or boiling point of water. On the Fahrenheit scale, the temperature at which water freezes is +32º. So 0º, the origin of this measurement, is very cold. On the Celsius scale, the freezing point of water is 0º, and +32º is quite warm. Both Fahrenheit and Celsius are relative scales, but because we understand their designated origins, they offer useful and reliable information.
On the Kelvin scale used by scientists, temperature measurement has a set origin, not a designated one. Absolute zero (0ºK) describes the point at which objects have the least amount of energy possible. Absolute zero is -273.15º Celsius and -459.67º Fahrenheit.
The system used to describe locations on Earth provides another example of how origins provide a common frame of reference. Think of the Earth as an enormous orange with 360 equal sections. The dividing lines between the sections are the meridians, which run between the North and South Poles. You can also think of meridians as halves of great circles encompassing the globe. These lines were arbitrarily established so we can determine longitude, a point's east-west measurement.
The "prime meridian," which passes through Greenwich, England, has been designated as 0º longitude. Longitudes are designated as E or W, that is, east or west of the prime meridian. The end line for measuring east and west longitudes is the other half of the prime meridian, the 180º line on the opposite side of the Earth.
If you drew a circle around your orange halfway between its top and bottom, this circle would correspond to Earth's equator. Unlike the prime meridian, the equator has a fixed location. For measurement purposes, the equator has been designated 0º latitude.
Above and below the equator are the latitude lines, called parallels because these concentric circles are parallel to the equator. Parallels help determine a point's latitude, or location along a north-south line. Latitudes north of the equator are labeled N; latitudes south of the equator are labeled S.
This complex system for finding locations on the Earth works because we all have agreed to accept the equator and the prime meridian as the origins for our measurements
Question: What is the purpose of identifying an origin? Answer: To provide a common frame of reference for describing a location or measurement.
Question: What is the prime meridian? Answer: The line of longitude that passes through Greenwich, England, designated as 0º longitude. | 677.169 | 1 |
Volume/47620: The Volume of a cube is given by V=s^3, where s is the length of a side. Find the length of a side of acube if the volume is 1000 cm^3 1 solutions Answer 31459 by Nate(3500) on 2006-07-29 22:03:22 (Show Source):
Graphs/47630: Graph the functions y = x and y =2 squared x on the same graph (by plotting points if necessary). Show the points of intersection of these two graphs. What are the points of intersection?
I also need to find an on-line graphing tool to graph the plots
Money_Word_Problems/47632: please help me solve this word problem.
Adult tickets for a play cost $20 and child tickets cost $16. If there were 27 people at a performance and the theater collected $464 from ticket sales, how many adults and how many children attended the play? 1 solutions Answer 31456 by Nate(3500) on 2006-07-29 21:57:30 (Show Source):
You can put this solution on YOUR website! adult tickets: a
child tickets: c
Combine:
+
=
Plug:
While eight adult tickets were sold, nineteen childeren's tickers were sold as well.
Trigonometry-basics/47633: Help please!- The approach pattern to an airport requires pilots to set an 11 degree angle of decent toward a runway.If a plane is flying at an altitude of 9,500 m at what distance (measured along the ground) from the airport must the pilot descend?-Thank you! 1 solutions Answer 31455 by Nate(3500) on 2006-07-29 21:53:58 (Show Source):
You can put this solution on YOUR website! This is, as you already know, a figure of a triangle.
Leg: 9500
Since you measure along the ground: leg: x
Angle: 11 degrees
tan(11/180) = 9500/x
x = 9500/tan(11/180)
About 2,835,140 m
Quadratic-relations-and-conic-sections/47606: This question is from textbook
Please assist us with this question..
In the following exercises, v denotes the vertex of a parabola, f the focus and d the directrix. Two of these are given. Find the third.
f(-2,5) v(-2,+1)
Question: How many points of intersection are there between the graphs of y = x and y = 2x²? Answer: There are two points of intersection. | 677.169 | 1 |
In the same way, we can use the Formulas (and our newly-christened right angle values) to explore the Second Quadrant: we simply throw our First Quadrant angles over the wall. For example,
This result makes some sense: The "vertical shadow" of a unit segment rotated to angle $\theta + 90^{\circ}$ matches the "horizontal shadow" of a unit segment rotated only to angle $\theta$; I bet it works the other way, too ... Hey, waydaminnit ...
At this point, we have two choices: retreat from the insanity, or embrace it. It turns out that the latter is the better course to take here: that single, tiny, intuition-shattering negative sign is the key to understanding how First Quadrant Trig extends to All Quadrants Trig.
You know the story from here: We use the Angle Addition Formulas to push from the Second Quadrant to the Third, to the Fourth, and beyond; and use the Angle Subtraction Formulas to assign trig values to negative angles. And we begin to make interesting observations that bolster our confidence in these values:
sine values are signed just like $y$ coordinates in each quadrant; cosine values just like $x$ coordinates; kinda convenient, that.
values repeat as we go all the way around the circle, because Angle Addition ultimately assures $\sin(\theta+360^{\circ}k) = \sin\theta$ and $\cos(\theta+360^{\circ}k) = \cos\theta$.
the triangle area formula $\frac{1}{2} a b \sin C$ now works for any-size $C$
Sure, we're forced to abandon the idea that the sine and cosine of an arbitrary $\theta$ should come from right triangles with angle $\theta$, but the gains from our expanded perspective more than make up for that. We all out-grow the training wheels sometime.
As I've admitted, I don't know how this conceptual progression matches with the actual history of trigonometry's development. However, I like using this approach as an object lesson in how mathematics often advances: we play around with intuitively-appealing notions, understand the heck out of that stuff by observing patterns, and let those observations guide exploration beyond our intuition's limits. The tail, as they say, wags the dog.
Of course, there are other engines driving mathematical advancement, too, but this seems to appeal to students. It helps makes the case that math is dynamic, subject to refinement (or overhaul) with every new discovery, and that its study is a never-ending and not-always-predictable journey.
Really nice answer, and from a teaching perspective is quite good. But as the you have said: "I don't know how this conceptual progression matches with the actual history of trigonometry's development". There seems to be a gap in history at that point, I can't find even a mathematician associated with that. – Luiz BorgesMay 16 '12 at 18:18
Question: What does the author mean by "throwing our First Quadrant angles over the wall"? Answer: The author means using the formulas to apply the trigonometric values from the first quadrant to the second quadrant by adding 90 degrees to the angle. | 677.169 | 1 |
In addition
to finding lines (axes) of symmetry, you can also look for points of symmetry.
A
point of symmetry is a point that represents a "center" of sorts for the figure.
For any line that you draw through the point of symmetry, if this line crosses the figure
on one side of the point, the line will also cross the figure on the other side of the
point, and at exactly the same distance from the point
For
instance, a figure-eight has a point of symmetry in the middle, where the lines cross (shown
by the blue dot):
For
an hyperbola, the center is the point of symmetry:
As
you can see from the hyperbola, a point of symmetry doesn't have to be a point on the figure;
it can lie outside the figure or graph; in this case, the point of symmetry happens to
be the origin.
You
can also view points of symmetry as being points about which you can rotate the shape 180°, as
shown below.
Axes
and points of symmetry can be anywhere on the plane. Points of symmetry do not have to be the origin;
lines of symmetry do not have to correspond to either axis.
point of symmetry: not
at the origin
line of symmetry: not along
either axis
The points and lines of symmetry do not even have
to touch the figure, and lines of symmetry do not have to be vertical:
point of symmetry: not
on the ellipse
horizontal line of symmetry: not
touching the hyperbola
It should be noted, however, that the lines
of symmetry for the ellipse and the vertical line of symmetry for the hyperbola will touch
their respective figures. Whether or not particular lines and points of symmetry will touch their
figures will vary from figure to figure.
Question: In an hyperbola, where is the point of symmetry located? Answer: The point of symmetry in a hyperbola is at the origin. | 677.169 | 1 |
Copy&paste 31.5n63.3w-32.7n62.9w, 32.7n62.9w-33.9n62.2w, 33.9n62.2w-35.5n60.6w, 35.5n60.6w-37.1n58.5w, noc, 35.5n60.6w-53.863n9.907w into the GreatCircleMapper for more info
Question: Which coordinate pair has the largest longitude value? Answer: 31.5n63.3w | 677.169 | 1 |
is designed for calculus students. Problem: you are given 100 feet of fence and
you are to enclose a figure that looks like a basketball key: consisting of a
rectangle with a semicircle attached to the top of the rectangle. Find the
dimensions of this shape that uses 100 feet of fence to enclose it and also has
the maximum area. Find that maximum area.
This activity is
designed for students to investigate how to calculate the distance from a point
to a line. Multiple representations are used: pencil and graph paper, graphing
calculator, CAS. Eventually the student will generate (derive) the Distance
From a Point to a Line formulas using CAS.
This activity uses the
Notes Q & A feature to simulate electronic flash cards. Right now there are
the trig unit circle values in both radian and degree modes, either from 0 to 2
pi or 0 to 360 degrees. More will be added later.
This activity uses the Notes Q & A feature to simulate
electronic flash cards. This is very similar to BG_1 except that the graphs
have been translated. There are 17 Basic Graphs, each on its own "card". Students
will be asked to state the equation that is graphed.
Question: What is the Notes Q & A feature used for in this activity? Answer: To simulate electronic flash cards | 677.169 | 1 |
Right Triangle Trigonometry: Real Life (non-linear) PowerPoint there are 6 examples of how to use right triangle trig to solve real life problems. These word problems include things like measuring sycamore trees, escalators in a mall, water slides, ski slopes, sightseeing in NYC, and creating a wheelchair ramp. These problems use sine, cosine, and tangent to solve for missing sides.
Presentation (Powerpoint) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1899
Question: How many real-life problem examples are there in the given PowerPoint presentation? Answer: 6 | 677.169 | 1 |
college cornerstone what time and financial constraints have you face since starting college? How did you deal with them
math The measure of the supplement of an angle is 20 degrees more than three times the measure of the original angle. Find the measures of the angles.
trig Two men on the same side of a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =120 m, find the distance (in meters) between the two men.
Question: What are the main topics discussed in the provided text? Answer: The text discusses time and financial constraints faced in college, the measure of supplementary angles, and a trigonometry problem involving the height of a building and the angles of elevation from two different points. | 677.169 | 1 |
Figures of Constant Width
At first thought, the circle would seem to be
the only two-dimensional figure of constant width. And as such, it can be used
as a wheel, without your vehicle bobbing up and down. But, it turns out that
there are infinitely many figures of constant width (depending on how you
define width). On the left we see two wheels of constant width (and an
animation on the right), and they too keep our vehicle (the thin rectangle on
top of our wheels) from bobbing up and down as they roll. It rolls as smooth as
can be (the two wheels don't have to be in synch with each other). The axle
does bob up and down, as you can see in the animation, so this wheel is of
limited practical value.
This figure is made up of three circular arcs of 60 degrees. It is the
second simplest of these figures of constant width (the circle being the
simplest). Such figures can be made from many circular arcs (the easiest to
construct), or with no circular arcs at all. The above diagrams were drawn with
the program Cinderella (the second one
with the help of Jasc's Animation Shop).
The above figure is called a Reuleaux triangle, and is also the shape of
the rotor in the Wankel rotory engine.
Question: What makes the wheels on the left roll as smooth as can be? Answer: They are made up of three circular arcs of 60 degrees each. | 677.169 | 1 |
Descriptive geometry used to be taught to engineers, not so often now that we have computer drawing software. The idea is to project 3D objects onto TWO half-planes, then flatten the half planes into a sheet of paper. There is a redundant dimension in the representation.
This can be exploited to visualize 4 dimensions: simply project a 4D object onto 2 planes. Descriptive geometry with both half-planes independent. One can go up to 6 dimensions by projecting onto 3 planes, or onto 2 volumes. The trick does help somewhat.
Question: What is the redundant dimension in the representation? Answer: The third dimension (depth) | 677.169 | 1 |
We know a polygon is a simple closed figure made up of only line segments. We can classify polygons according to the number of sides or vertices.
The simple polygon we know is a triangle. A triangle has three sides and, thus, is a three-sided polygon.
A four-sided polygon is called a quadrilateral.
A five sided polygon is called a pentagon.
In this manner, we can obtain a six-sided polygon called a hexagon, a seven-sided polygon, called a heptagon, and so on.
If we have an n-sided polygon, it is called an n-gon.
CONCAVE & CONVEX POLYGONS:
We know that each side of a polygon is connected by two consecutive vertices of the polygon.
A diagonal is a line segment that connects the non-consecutive vertices of a polygon.
If a diagonal lies outside a polygon, then the polygon is called a concave polygon.
If all the diagonals lie inside the polygon, then the polygon is said to be a convex polygon.
REGULAR & IRREGULAR POLYGONS:
A regular polygon is equiangular and equilateral. The word equiangular means, the interior angles of the polygon are equal to one another. The word equilateral means, the lengths of the sides are equal to one another.
The polygon with unequal sides and unequal angles is called an irregular polygon.
ANGLE SUM PROPERTY:
The sum of all interior angles of a polygon is called the angle sum. , .
At one vertex, we extend a side. This side makes an angle with its consecutive side. This angle is called the exterior angle. The interior angle and the exterior angles are adjacent angles. These angles form a linear pair. Hence the sum of the exterior angles of any polygon is
Question: What is the sum of the exterior angles of any polygon? Answer: 360 degrees
Question: What is the minimum number of sides a polygon can have? Answer: 3 (A triangle) | 677.169 | 1 |
Some shapes are pleasing because they are so symmetrical and other shapes are pleasing because they are so complicated.
Shapes are the concern of the area of mathematics called geometry. However, here we will adopt a broader view of geometry, which can be thought of as the science of studying visual patterns. Shapes are only one part of the story.
Careful Looking
Take a careful look at the shape below, which you should think of as being a drawing in the plane (flat surface) of an object in 3 dimensions.
Figure 3
Do now 1:
a. Write down on a piece of paper a list of as many properties or facts about the object (Figure 3) as you can. (Minimum 5)
b. Exchange your list with the person sitting next to you. Use the list of your neighbor to help you add 5 new additional properties to your original list.
When one looks at shapes and wants to describe them to other people, one uses words. Some of the words that might be used in describing the object in Figure 1 are:
Immediately you may realize that different people may use different words for the same thing. Also different people may use the same word but have different meanings in mind - and I will restrict myself to only using words in English. Furthermore, sometimes words that we use in common everyday language may be used in a different way from the way that these terms are used in mathematics.
Exercise 1
Practice careful looking at these objects:
a.
Figure 4
Figure 5
Here I will try to show that historically the evolution of the meaning of words has helped drive the creation of new mathematics. For example, all of the diagrams in Figure 6 would today be called polygons, but at times in the past this would not have been true!
Figure 6
Figure 6 illustrates convex 3-gons and 4-gons and a non-convex polygon with 12 sides, as well as a self-intersecting polygon with 6 sides. Sometimes it will be convenient to think of polygons as points (vertices) joined by rods, and sometimes as filled in regions with a certain number of sides. I will refer to these as rod and membrane models for the polygon concept. When one thinks of a polygon as a system of "rods," it is still customary to refer to a polygon such as the 4-gon in Figure 6 as convex. The usual definition of a convex set is that a set X is convex if for any two points p and q in X the line segment joining p and q is also in X. Strictly speaking, if one is thinking of the 4-gon in Figure 6 as a collection of rods, one should say that this polygon together with its interior points is convex.
One of the most fundamental properties of the Euclidean plane is that if one draws a polygon which does not intersect itself such as the one in Figure 7, then the polygon divides the plane into three sets: those points on the polygon, those points in the interior of the polygon, and those points in the exterior of the polygon.
Figure 7
Question: Which of the following is NOT a part of the object in Figure 3, according to the text? A) Edges B) Vertices C) Faces D) Dimensions Answer: D) Dimensions.
Question: What is the minimum number of properties or facts about the object in Figure 3 that the text asks to list? Answer: 5. | 677.169 | 1 |
Figure 11 is a rhombus, trapezoid, parallelogram, rectangle, kite, and square. In this situation we can use the definitions of the words to determine that Figure 11 is an example of all of these types of quadrilaterals. However, there are quadrilaterals which are, for example, rhombuses but not squares.
Furthermore, thought of as a graph, Figure 9 and Figure 11 show the "same" graph. Both of these figures have 4 vertices and 4 edges. The technical term for when two graphs look different but we can treat them as if they are the same is isomorphic. While Figure 12 is not a polygon (it has some curved sections), as a graph it is isomorphic to the the graphs in Figures 9 and 11. The word "isomorphic" here is designed to connote that one has two things but they have the "same structure." Mathematicians talk about isomorphic graphs, groups, rings, and fields. The term comes up in geometry, topology, and algebra as well as many other parts of mathematics. The formal definition of isomorphic is in terms of the concept of a function called an isomorphism. Isomorphisms preserve some feature of the objects they relate. For graphs, they preserve the way that vertices are joined up to each other by the edges.
Figure 12
Figure 13 shows another graph with 4 vertices and 4 edges but it is not isomorphic to the graphs in Figures 9, 11, 12.
Figure 13
If graphs have the same structure it seems natural that their essential properties would be the same. Thus, isomorphic graphs would have the same number of vertices and edges. They would also have the same numbers occurring as the valences of the vertices of the graphs. Figure 8 has vertices of valence 2, 2, 2, 2. Figure 13 has vertices of valence 3, 2, 2, 1. Hence, the graphs in these figures are not isomorphic. Having the same valences does not, however, guarantee that there is an isomorphism between two graphs. All three of the graphs in Figure 14 look different. However, they all have valences 4, 4, 2, 2, 2, 2,
Figure 14
Two of the graphs in Figure 14 are isomorphic to each other but not to the third graph. Can you tell which two graphs are isomorphic? The two graphs in Figure 14 which are isomorphic as graphs have a sense in which they are not the same. Whenever, one has items which can be distinguished from each other because of additional attributes that were not previously contemplated, one has made progress.
Question: What are the valences of the vertices in Figure 13? Answer: The vertices in Figure 13 have valences 3, 2, 2, 1.
Question: Are Figures 9 and 11 the same graph? Answer: Yes, Figures 9 and 11 are isomorphic, meaning they have the same structure. | 677.169 | 1 |
So take r3=0 and you have solved the problem. Additionally, Dots can be named circles with r=0 so C3 is still a perfectly fine circle. The other intersection is at (0.5;-sqrt(3/4)) so C3 should have r3=2sqrt(3/4) to join in that intersection but then r3 is no integer)
Taking n>1, we'll get the y-coordinate of the intersection of C1 and C2 to be at sqrt(n^2-1/4). This will be 15/4, 35/4, 63/4, 99/4, 143/4,... for n=2,3,4,5,6... These will never give an integer as coordinate.
That means that the radius of C3 will be (y-coordinate minus sqrt(3/4)) or (minus y-coordinate + 2sqrt(3/4)) The latter for the intersection in the negative area. Also these will never give an integer.
That means that the only solution for r1=r2 will be (r1,r2,r3)=(1,1,0)
It gets a lot more complicated for having r1 != r2, but here you can still work with symmetry because your figure is symmetrical.
If r1 > r2, the intersections will move to the right, but also the y-coordinate will change (it will increase first - positive region- , as r1 increases, but then it'll decrease - the y-coordinate will be zero if r1=r2+1 leaving only 1 intersection).
If r1 < r2 then you'll have exactly the same, but with the intersections moving to the left.
The left one (red dot at the left side) is at (0,0) The right one is at (1,0) The middle one is at (0.5;sqrt(3/4))
We have to find three radii (they may all be different) that will let the three circles intersect at a certain point (The black blob is where the three intersect in my image). That's easily done (just make the picture).
The added restriction is to let all radii be of an integer value. Then there is only one solution: r1=r2=1, r3=0. They will all intersect at (0.5;sqrt(3/4))
Okay, that's the problem I solved in my first post as well and I came to pretty much the same conclusion, except I didn't consider a radius r3=0 as a legitimate solution. I just considered the kind of points where C1 and C2 can possibly intersect and the kind of points where C1 and C3 can intersect and couldn't find any overlap there. Apparently this didn't satisfy DeadMG, so I'm not quite sure where he thinks the problem lies.
_________________ They often call Latin a dead language, but it would be more accurate to call it a zombie language, infecting everyone who comes too close to it.
Question: What is the only solution for r1=r2 where all radii are integers? Answer: (r1,r2,r3)=(1,1,0)
Question: What is the y-coordinate of the intersection of C1 and C2 for n=6? Answer: 143/4
Question: Where are the intersections of the three circles when r1=r2=1, r3=0? Answer: At (0.5;sqrt(3/4))
Question: What happens to the y-coordinate of the intersection of C1 and C2 as r1 increases? Answer: It increases first, then decreases | 677.169 | 1 |
Center of a graph: The center of a graph G consists of the subgraph of G induced by the set of vertices of G with minimal eccentricity. (The eccentricity of a vertex v is the distance v has from the vertices that are farthest (graph distance) from it.
Median of a graph: The median of a graph G consists of the subgraph of G induced by those vertices which minimize the sum of the distances to all the other vertices of the graph.
Theorem:
The center of a tree consists either of a single vertex or a single edge.
Theorem:
The median of a tree consists of either of a single vertex or a single edge.
Both of these theorems are due to the topologist Camille Jordan (1838-1922), for whom the Jordan Curve Theorem is named.
Question: What does the median of a graph consist of? Answer: The median of a graph G consists of the subgraph induced by those vertices which minimize the sum of the distances to all the other vertices of the graph. | 677.169 | 1 |
I think that the accuracy of our design turned out to be low, unfortunately. This is because the tape that we used to mark the angles, kept moving around due to the wet pavement. Sometimes, people accidentally stepped on the tape markers and we had to relocate where the marker should be (if we didn't, further angles and measurements would be way off the track as well). Although we didn't think of this on the day, we could have had 1 person recheck all of the angles using our homemade compass to verify our accuracy. This would have been efficient because 2 of us were usually just hanging around because some operations only required 1 or 2 people. Also, it is really common for us to occasionally confuse ourselves on how to bisect angles, so redoing them, just in case, would definitely have made our sandbox more accurate.
Of the 3 angles we measured, 2 of the angles were extremely accurate (they were exactly 90-degrees). These 2 angles happened to be 90-degree angles. The reason that they were more accurate than the other angle (a 45-degree angle) is that we only had to do one operation. To find a 90-degree angle, you only need to use the perpendicular bisector. However, to find a 45-degree angle, you have to first use the perpendicular bisector to find a 90-degree angle and then use the angle bisector to split a 90-degree angle in half, resulting in a 45-degree angle; a bit more complicated. The 45-degree angle we tried to recreate turned out to be 5 degrees off; a 50-degree angle. If we had completed the shape, the unknown side would probably be quite a bit off because of the inaccuracy of the 2nd angle we did (the 45-degree angle), as it would progressively affect the following measurements we did.
When trying to measure angles with a tiny protractor, it was very difficult because the lines on the protractor are very close together, and you can't be accurate in which degree your angle is. This is because we used 2-centimeter wide tape to create our shape and the degree lines in the protractor were only approx. 1 millimeter wide. This is problematic because inaccurate angles will change the length of the last side (since the last side MUST complete and connect the shape).
Question: What made measuring a 45-degree angle more complicated than a 90-degree angle? Answer: To find a 45-degree angle, they had to first find a 90-degree angle and then split it in half, while a 90-degree angle only required using the perpendicular bisector.
Question: What was the main issue that led to the low accuracy of their design? Answer: The tape used to mark angles kept moving around due to wet pavement and people accidentally stepping on the tape markers. | 677.169 | 1 |
1. CONSTRUCT A COMPLETE UNIT CIRCLE: this shall include all "30,45,60" degree angles in each quadrant, as well as the axis angles. In addition to drawing each angle, you must label all angles with both their degree and radian measure. Additionally, you must also write the coordinates for each corresponding point along the unit circle. *NOTE* Use your book to help you with this; this is supposed to be the easy part of the assignment.
2. COMPUTE THE SIX TRIGONOMETRIC FUNCTIONS FOR EACH ANGLE ON THE UNIT CIRCLE: for example, start with 0 and compute sin(0), cos(0), tan(0), csc(0), sec(0), and cot(0); then move to pi/6 and compute sin(pi/6), cos(pi/6), and so on. Again, you will have to find all six trig ratios for EACH angle on the unit circle. We discussed how you can find these ratios from the unit circle in class today.
Question: What is the radian measure of this angle? Answer: 0 radians | 677.169 | 1 |
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Two lines are said to be parallel, if they are in the same distance away at each point.
That is, two lines are said to be parallel if
They both lie in the same plane and
They do not intersect (or cross each other)
The above definition really means that parallel lines never cross each other.
What is a transversal?
A line that crosses a pair of parallel lines on a slant is called as the transversal line. Totally eight angles are formed as the transversal line crosses the parallel lines. In the 8 angles, one can observe that 4 of them are quite large whereas remaining 4 of them are quite small.
A very important set of properties of these angles is that the angles which appear to be same really are exactly the same. Therefore,
Question: Are the angles on the same side of the transversal equal? Answer: Yes | 677.169 | 1 |
Question 379437 Click here to see answer by robertb(4012)
Question 379 Click here to see answer by mananth(12270)
Question 379922: a piece of wire 10m long is cut into two pieces. one piece is bent into a square and the other is bent into an equilateral triangle. how should the wire be cut so that the total area enclosed is a)a maximum b) a minimum Click here to see answer by richard1234(5390)
Question 380410: I saw the answer to my question but did not really understand how it was broken down. Why is the answer 95 degrees to this question.
The second angle of a triangular building lot is three times as large as the first. The third angle is 10degrees more than the sum of other two angles. Find the measure of the thrid angle. Click here to see answer by richard1234(5390)
Question 380483: total revenue is the total amount of money taken in by a business. an appliance firm determines that when it sells x washing machines the total revenue, R, in dollars, is given by the polynomial R=239.46x-0.1x^2. what is the total revenue from the sale of 165 washers Click here to see answer by mananth(12270)
Question 380509: help please and thank you
A launched rocket has an altitude, in meters, given by the polynomial h+vt-4.9t^2, where h is the height, in meters, from which the launch occurs, at velocity v in meters per second, and t is the number of seconds for which the rocket is airborne. If a rocket is launched from the top of a tower 70 meters high with an initial upward speed of 80 meters per second, what will its height be after 2 seconds?
Question 381354: I need a break down of this problem please. Not really good at word problems.
The property restoration company PuroServ is considering switching to new dehumidifiers. Their market research, considering the cost of the new machines and their efficiency, tells them that the switch would give them an 80% chance of making a $20,000 profit, a 15% chance of breaking even, and a 5% chance of losing $5,000. How much money does PuroServ expect to make with their new purchase? Click here to see answer by robertb(4012)
Question: What is the total length of the wire used in the first problem? Answer: 10 meters | 677.169 | 1 |
direction I am pointing, represented by three angles - one for each angle of rotation (rotation in X, rotation in Y, rotation in Z) (for the sake of the example let's assume I'm one of those old logo turtles with a pen) and the distance I will travel in the direction I am pointing.
What does 'rotation in x, rotation in y' mean? You can't rotate in one dimension - in two dimensions there's only 1 plane of rotation, and in 3D there are just 2 planes of rotation. – Kirk BroadhurstJan 10 '11 at 0:28
@Kirk is right. Think this way: for representing a point in 3D you need 3 scalars (numbers). But you have 4 (3 angles and a distance). Something is superfluous ... – belisariusJan 10 '11 at 0:34
3 Answers
Based in the three angles you have to construct the 3x3 rotation matrix. Then each column of the matrix represents the local x, y and z directions. If you have a local direction you want to move by, then multiply the 3x3 rotation with the direction vector to get the result in global coordinates.
I made a little intro to 3D coordinate transformations that I think will answer your question.
3D Coordinates
First, it is strange to have three angles to represent the direction -- two would be enough. Second, the result depends on the order in which you turn about the respective axes. Rotations about different axes do not commute.
Question: Why are three angles used to represent a direction in 3D space, when only two would be sufficient? Answer: The text suggests that this is unusual and not typically done. In 3D space, two angles (like azimuth and elevation) are usually sufficient to represent a direction.
Question: What does it mean when the text says "rotations about different axes do not commute"? Answer: This means that the order in which rotations around different axes are performed matters. The result of performing rotations in one order may be different from performing them in a different order. | 677.169 | 1 |
Circle
called, centre, qv, diameter, ratio, circles, plane and geometry
CIRCLE (from Lat. cireulus, dim. of circus, Gk. dpKos, rk og. Kpkos, kriko.c, circle). The locus (q.v.) of all points in it plane at an equal finite distance from a fixed point in that plane. The fixed point is called the centre, and the spate inelosed. Or, more properly, its
ure, the area of the circle. The segment of any straight line intercepted by the circle (AR in Fig. 11 is called a chord. Any chord passing
the centre. 0, is called a diameter. as _VW. The centre bisects any diameter, and the halves are called radii. Any line drawn from an external point cutting the circle. as PQ, is called a secant : and any line which has contact with the circle, but does not intersect it when produced. as 11'1'. is called a tattrent. Any por tion of the area limited by two ra dii. as OA and Olt, is called a sector: and any port ion of the circle, 11A'A, is called an are. A chord is said . to divide the area into segments: the segments are equal if the chord is a diameter. A plane passing through the centre of a sphere cuts the surface in a circle called a great cir cle of the sphere. Circles of longitude are great circles. Other circles of a sphere are called small circles. .Ancient writers usually called the circle, as above defined, a circumference, the word `circle' being applied to the space inclosed. In modern geometry, at least alcove the elements, the word 'circumference' is not used, and the word 'circle' applies to the curve.
Question: What is the part of a circle that lies between two radii called? Answer: A sector | 677.169 | 1 |
Cut The Knot!
Inversion
'I suppose I must have gone round in a circle.'
The sergeant again exchanged a knowing glance with the whole personnel of the station. 'A fine circle, that circle of yours!'
Jaroslav Hasek The Good Soldier Svejk, Penguin Books, 1983, p. 255
Let's consider the following problem that, perhaps surprisingly (because of its simplicity), has several apparently unrelated solutions. Which one sheds more light on the nature of the problem?
The configuration consists of a circle with center O and a straight line ST that cuts from a circular segment STS. Circles are inscribed in the segment and, for each, the points A and B of tangency with the segment are joined by a straight line. Prove that all those lines concur at the midpoint M of the arc ST complementary to the segment.
Proof 1
Let O' be the center of an inscribed circle. Then AO'B is isosceles. Extend AB beyond A and let it intersect the perpendicular OM to ST at point N (not shown on the diagram.) The two triangles AO'B and NOB are similar. Indeed, they have a common angle at B and, since ON||O'A, their respective angles at O' and O are also equal. NOB is therefore isosceles. OB = ON, which implies N = M.
The next proof builds on the observation that the triangles AO'B and MOB are not merely similar but are homothetic with center B.
Proof 2
All circles are similar and, moreover, homothetic. For each pair of distinct circles there are either two or one homothety that maps one of the circles onto the other. Two touching circles (as in the problem) are related by a single homothety with their common point of tangency as the center. All points related by a homothety are collinear with its center B. In particular this is true of the lowest (A and M in the diagram) points of the two circles.
Proofs 1 and 2 are simple and are not exactly ad hoc as both apply to a more general situation of a line and a circle that do not necessarily cross and to the circles tangent to both of them.
Proof 3
Let's make an inversion with center S and radius SM. The point M remains unmoved as a point on the circle of inversion. The line ST is fixed, although not point-wise. The circle becomes a straight line meeting ST at the image T' of the point T and passing through M. Since inversion preserves angles, the tangency points A and B become the tangency points A' and B' in the inverse image. The circles inscribed in the segment map onto the circles inscribed into the angle vertical to the angle ST'M.
Question: What is the year of publication of the book? Answer: 1983
Question: What does the inversion in Proof 3 transform the circle into? Answer: A straight line
Question: Who is the author of the book "The Good Soldier Svejk"? Answer: Jaroslav Hašek | 677.169 | 1 |
Polar Representation of Complex Numbers
The Argand diagram
In two dimensional Cartesian coordinates (x,y), we are used to plotting the function
y(x) with y on the vertical axis and x on the horizontal axis.
In an Argand diagram, the complex number z = x + iy is plotted as a single point with coordinates (x,y). The horizontal axis is called the real axis (x-axis) and the vertical axis is called the imaginary axis (y-axis).
As in usual Cartesian coordinates, the distance from the origin to a point
(x, y) is equal to
. This is equal to the modulus
| z | of the complex number
z = x + iy.
The Argand diagram may also be called the complex plane. It stresses that complex numbers are a generalisation of real numbers, that lie on the horizontal axis only.
The expression z = x + iy is known as the Cartesian form or the rectangular form of the complex number z. Using the Argand diagram, we can see that the addition of complex numbers behaves like the addition of vectors.
If we express z = x + iy as an ordered pair (x, y), then the addition of two complex numbers may be defined by
in the same way as the addition of two vectors.
Polar coordinates
A position vector of a point in two dimensions may be expressed in terms of Cartesian coordinates (x,y) and plotted with y on the vertical axis and x on the horizontal axis.
It is also possible to express the two dimensional position vector in terms of polar coordinates (r,θ) where r is the magnitude of the vector (distance from origin to the point) and θ is the angle between the position vector and the positive x-axis.
The Cartesian and polar coordinates are related by:
In the same way, the complex number z = x + iy may be expressed in polar coordinates (r,
θ), in its polar form:
where
The modulus of a complex number
In polar coordinates (r, θ) the magnitude r of the distance from the origin to the point represented by
z is equal to the modulus of the complex number |
z |:
The argument of a complex number
In polar coordinates (r, θ) the angle θ is known as the argument of the complex number
z, denoted θ = arg(z).
There is a complication because a single point on the Argand diagram does not correspond to a single complex number. The reason is that we can add
2π to the value of the argument θ in order to produce a different complex number, but when plotted on the Argand diagram, the two numbers are plotted in the same place.
Principal value: If we want to uniquely define the value of the argument θ we can impose the condition
−π<θ≤π so that θ is known as the principal value of the argument.
For the complex number z = x + iy, the argument θ is given by the solution of the equations:
or
If the second expression tan θ = y/x is used to determine θ, it is wise to plot
z = x + iy on an Argand diagram to check that the answer is correct.
Question: What is the relationship between the Cartesian and polar coordinates of a complex number? Answer: They are related by the equations x = r cos(θ) and y = r sin(θ).
Question: What is the expression z = x + iy known as? Answer: The Cartesian form or the rectangular form of the complex number z.
Question: What is the horizontal axis called in an Argand diagram? Answer: The real axis (x-axis)
Question: How is the distance from the origin to a point (x, y) in an Argand diagram represented? Answer: It is equal to the modulus |z| of the complex number z = x + iy. | 677.169 | 1 |
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Symmetry
Symmetry is when the beatmap is symmetrical in respect to an axis. The most common type of symmetry is horizontal symmetry. There are other types of symmetry too, however, like vertical symmetry, diagonal symmetry, and symmetry with respect to the origin (Normally we would say, "Think of an odd function", but that would probably alienate a large portion of the playerbase).
To guide you through the process of understanding symmetry, we are going to go through examples of symmetry in a map that is already ranked so you understand how it works. Trying to explain symmetry with just text would be futile and overall not very effective. Here is the example map we will be using (we'll be looking at the Expert difficulty; not a stellar map by any means, but it will do):
Contents
Horizontal Symmetry
This map starts off with an early example of horizontal symmetry at 00:03:655 (1,2,3,4,5,6,7,8). This was done by first figuring out the combo we wanted to map to. In this case it was a simple combo of 8 beats. This meant that there were two easy ways to create symmetry here:
1. Have four beats represented on each side of the y-axis.
2. Have beats 1 and 5 on the y-axis, with 2,3, and 4 on one side and 6, 7, and 8 on the other side.
Here, #2 was clearly chosen. To make sure the symmetry was effective, hit circles (2,3,4) were copied and pasted where we wanted (6,7,8) to be. Then, (6,7,8) had a Reverse Selection applied to them in order to keep the spacing correct. Some newer mappers make the mistake of just blindly copying and pasting, without modifying things as needed to keep the map's flow up. This is a bad habit and is not recommended.
Another, much more complex example of horizontal symmetry is at 00:29:455 (1,2,3,4,5,6,7,8,9). First, (1,2,3) is a simple smiley face pattern; 1 and 2 were just placed an equal distance apart from the y-axis, with a symmetrical slider placed beneath them. It then gets a bit more complex, though.
The horziontal symmetry is maintained in the rest of the combo; just see for yourself. These types of patterns typically involve a lot of experimenting to get just right. Here you'll notice that the spacing was almost perfectly maintained; a jump was not wanted here, so keeping the spacing correct as a necessity. Doing this can take a while, but is often well worth it. Also, you'll notice that (6) is technically not symmetrical. However, having it there makes the pattern funner and flow better without destroying the pattern, so in it goes.
You'll notice more examples of horizontal symmetry in this and other maps. Just experiment and see what works for you.
Vertical Symmetry
Question: What is the second, more complex example of horizontal symmetry in the example map? Answer: At 00:29:455 (1,2,3,4,5,6,7,8,9)
Question: What is the benefit of maintaining correct spacing in a symmetrical pattern? Answer: It prevents unwanted jumps and improves the flow of the map
Question: In the example map, what is the first instance of horizontal symmetry? Answer: At 00:03:655 (1,2,3,4,5,6,7,8)
Question: What is the most common type of symmetry in a beatmap? Answer: Horizontal symmetry | 677.169 | 1 |
Distance formula Teacher Resources
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Middle and high schoolers solve 14 problems that include finding the distance between two points in different situations. First, they refer to a number line to find the measure of each point on the line. Then, pupils refer to the coordinate plane shown to find each measure and round the answers to the nearest hundredth.
Pupils calculate the distance using the correct formula. In this geometry activity, students apply concepts of the Pythagorean Theorem and right triangles to solve problems. They solve problems involving conics and their distance or circumference.
In this distance formula worksheet, students solve and complete 28 various types of problems. First, they find the distance between each pair of points provided. Then, students solve and graph the points.
Students watch teacher demonstration of distance formula, view non-collinear cities on maps, plot points on coordinate plane, evaluate square roots, compute distance between two points, and use Heron's formula to find area of triangles.
Using a longitude and latitude map that relates to the coordinate grid and job description cards that come with places to travel to, young mathematical travelers use the distance formula and The Pythagorean Theorem to answer questions and report on their travels and overall cost. Students use specific websites to gather information.
For this online math worksheet, high schoolers practice their ability to calculate the distance between two points by utilizing the distance formula. Students can check their answers, and there are "hints" given should they get stuck.
Pupils determine the distance between two given points. They use the distance formula to determine line length, as well as the length of the radius of a circle. The Pythagorean Theorem may also be used to determine line length. This one-page worksheet contains ten problems.
Question: What is the first step students take before calculating the distance between two points? Answer: They find the measure of each point on the line using a number line.
Question: Which of the following is NOT a part of the worksheet? A) Solving and graphing points B) Finding the distance between two points C) Writing an essay D) Checking answers Answer: C) Writing an essay | 677.169 | 1 |
How would one best describe an ellipse?
Question
#62330. Asked by minuscule_. (Feb 06 06 8:32 PM)
smeogalla
a regular oval resulting when a cone is cut obliquely by a plane. from the greek elleipsis- deficit
Feb 06 06, 11:03 PM
romeomikegolf
a closed plane curve resulting from the intersection of a circular cone and a plane cutting completely through it; "the sums of the distances from the foci to any point on an ellipse is constant" [syn: oval]
Or, in plain english, the shape of a rugby ball.
Feb 06 06, 11:06 PM
gmackematix
Well, a cross-section of a rugby ball.
It can best be described with two pins and a loop of string.
Or a pair of compasses and a piece of paper wrapped around a cylinder.
Or with a computer and the formula y = b^2.sq rt (1 - x/(a^2)) where a and b are the major and minor axes of the ellipse.
Feb 07 06, 1:30 AM
gmackematix
Even more simple methods include using an elliptical stencil, clicking the circles button in a MS Office program and dragging the mouse or finally, doing nothing and letting your planet describe an ellipse for you.
Feb 07 06, 12
Question: What is the relationship between an ellipse and a cone? Answer: An ellipse is the result of a circular cone being cut by a plane.
Question: What is the Greek origin of the word 'ellipse'? Answer: elleipsis- deficit | 677.169 | 1 |
You can put this solution on YOUR website! it is difficult to know what your asking here!!! SAS -side,angle,side- is a theorem which states that if any two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle the triangles are congrent. You will have to be more specific what your looking for if this doesnt help
You can put this solution on YOUR website! if you want to prove that two triangles are congruent by SAS, you have to show that two corresponding sides are congruent and that the angle between those corresponding sides are congruent also.
---
once you do that, the triangles are proven congruent by SAS.
---
for example:
suppose you are given an isosceles triangle ABC
AB and BC are the equal legs.
BC is the base.
you are then told that AD is the angle bisector of angle ABC.
you might then be asked to prove triangle ABD is congruent to triangle CBD
----
you would start by saying:
AB congruent to BC (given)
BD congruent to BD (same line)
angle ABD is congruent to angle CBD (bisector of an angle creates two congruent angles with a common side of each of these angles being the bisector.
triangle ABD is congruent to triangle CBD by SAS
---
that's your proof.
Question: What is the purpose of AD in the example? Answer: AD is the angle bisector of angle ABC | 677.169 | 1 |
Laying out an Ellipse
Kevin Boyle explains how to properly lay out an ellipse
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Mon, 25 Jun 2012|
Transcript
Laying out circles in the shop is an easy task when you've got your compass, a centerpoint—strike a circle no problem. But laying out an ellipsis is a little trickier. Today we're going to show you a way of doing that with a framing square and a shopmade trammel. Now what we're going to start with is basically laying out the size of your ellipse: The two main factors are your major axis and your minor axis. (Major axis being the length of your ellipse, minor axis being the width). I've laid those out; now it's time to make a trammel. A trammel is made of scrap stock that is roughly 3 inches longer than half your major axis. Start by drilling a hole one inch from one end that fits your pencil snugly. Drive two 4-penny finish nails into the stock, one is half the major axis, one is half the minor axis. The other thing you want to do is file down the tips of the nails after you've driven them through. Okay, now we're going to lay out the arc in our first quadrant. To do that, I'm using a framing square. I've applied two pieces of double-faced tape to the backside as to kinda help hold this in position. So we lay that down, line that up with our major and minor axis. Just kinda press that in place. Take our shopmade trammel, line up the finishing nail with the corner of the framing square, and when we pull this around, we want to keep both nails in contact with the framing square, and we're just going to strike that arc. There's our ellipse. We've laid out all four quadrants. Remove our framing square; you've got a perfect ellipse.
Question: What should the length of the scrap stock used to make the trammel be? Answer: Roughly 3 inches longer than half the major axis of the ellipse. | 677.169 | 1 |
word "secant" comes from the Latin word meaning "cutting."
Similarly, the cosecant of the angle AOB is the line OG from the center of the circle
to the cotangent line FG.
Exercises
Note: as usual, in all exercises on right triangles c stands for the hypotenuse,
a and b for the perpendicular sides, and A and B for the angles opposite
to a and b respectively.
26. In each of the following right triangles of which two sides are given, compute the sin, cos,
and tan of the angles A and B. Express the results as common fractions.
(i). c = 41, a = 9.
(ii). c = 37, a = 35.
(iii). a = 24, b = 7.
31. In a right triangle c = 6 feet 3 inches and
tan B = 1.2. Find a and b.
34. a = 1.2, b = 2.3. Find A and c.
42. a = 10.11, b = 5.14. Find B and c.
In the next few problems, the triangles aren't right triangles, but you can solve them using
what you know about right triangles.
61. In an oblique triangle ABC,A = 30°, B = 45°,
and the perpendicular from C to AB is 12 inches long. Find the length of AB.
67. If the side of an equilateral triangle is a, find the altitude, and the radii of the
circumscribed and inscribed circles.
202. From the top of a building 50 feet high the angles of elevation and depression of the top and
bottom of another building are 19° 41' and 26° 34', respectively. What are the height and
distance of the second building.
207. From the top of a lighthouse 175 feet high the angles of depression of the top and bottom of
a flagpole are 23° 17' and 42° 38', respectively. How tall is the pole?
214. At two points 65 feet apart on the same side of a tree and in line with it, the angles of
elevation of the top of the tree are 21° 19' and 16° 20'. Find the height of the tree.
215. As a balloon passes between two points A and B, 2 miles apart, the angles of
elevation of the balloon at these points are 27° 19' and 41° 45', respectively. Find the
altitude of the balloon. Take A and B at the same level.
233. The top of a lighthouse is 230 feet above the sea. How far away is an object which is just
Question: What is the origin of the word "secant"? Answer: The word "secant" comes from the Latin word meaning "cutting."
Question: In the first exercise (i), what is the length of the hypotenuse (c)? Answer: 41
Question: In problem 61, what is the measure of angle A? Answer: 30°
Question: In exercise 42, what is the length of 'b'? Answer: 5.14 | 677.169 | 1 |
In this example, each of the roads leading up to the intersection is one block
long. We found earlier that the angles opposite each other in the intersection
have equal measure due to the Vertical Angle Theorem. Since the
sides have equal length and the included angles are the same, the two triangles
formed, Triangle ABC and Triangle EDC are congruent
by SAS. If you think about the Parallel Line Property,
congruency could also be proved by AAS and ASA...
try it!
Note that there is a footpath that extends from vertex C down
to segment DE (in the colored diagram). This footpath actually splits vertex C
into two equal angles. This would then be called an angle bisector.
We also note that in this case the footpath bisects segment DE
at a perpendicular angle. Consequently, the footpath could also be known as a
perpendicular bisector.
The Linear Pair Property (LPP) is shown above as well. The
LPP simply refers to a line that is intersected by another line and creates
two angles that add up to 180 degrees. For example, angles 6 and 7
form a linear pair.
Question: How many triangles are formed at the intersection? Answer: Two | 677.169 | 1 |
Yes, the Pythagorean Theorem is involved in proving this stuff. Yes, these are the same letters
used in the Pythagorean Theorem. No, this is not the same as the Pythagorean Theorem. Yes,
this is very confusing. Accept it, make sure to memorize the relationship before the next test,
and move on.)
For a taller-than-wide ellipse with center at (h, k),
having vertices a units above and below the center and foci c units above and below the center, the ellipse equation is:
An ellipse equation, in conics form, is always "=1". Note that, in both
equations above, the h always stayed with the x and the k always stayed with the y. The only thing that changed between the two equations was the placement
of the a2 and the b2. The a2 always goes with the variable whose axis parallels the wider direction of
the ellipse; the b2 always goes with the variable whose axis parallels the narrower direction. Looking at
the equations the other way, the larger denominator always gives you the value of a2, the smaller denominator always
gives you the value of b2, and the two denominators together allow you to find the value of c2 and the
orientation of the ellipse.
Ellipses are, by their nature, not "perfectly round" in
the technical sense that circles are round. The measure of the amount by which an ellipse is "squished"
away from being perfectly round is called the ellipse's "eccentricity", and the value
of an ellipse's eccentricity is denoted as e
= c/a. Since the foci are closer to the center
than are the vertices, then c < a, so the value of e will always be less than 1. If an ellipse's foci are pulled inward toward the center, the ellipse will
get progressively closer to being a circle. Continuing that process, if we let c = 0 (so the foci are actually at the center),
this would correspond to e = 0, with the ellipse really being a circle. This tells us that the value of e for a true (non-circle)
ellipse will always be more than 0. Putting this together, we see that 0 <
e < 1 for any ellipse.
When scientists refer to something (such as Pluto) as having an
"eccentric" orbit, they don't mean that the orbit is "weird"; they mean that
it's "far from being circular". In Pluto's case, its orbit actually crosses inside that
of Neptune from time to time. The larger the value of e, the more "squished" the ellipse.
A physical property of ellipses is that sound or light rays emanating
from one focus will reflect back to the other focus. This property can be used, for instance, in
medicine. A patient suffering from, say, gall stones can be placed next to a machine that emits
shock waves away from the patient and into an elliptical bowl. The patient is carefully positioned
Question: What does it mean when scientists refer to an object's orbit as 'eccentric'? Answer: They mean that the orbit is far from being circular.
Question: What is a unique property of ellipses? Answer: A unique property of ellipses is that sound or light rays emanating from one focus will reflect back to the other focus.
Question: What happens to the value of 'e' as the foci of an ellipse are pulled inward toward the center? Answer: As the foci are pulled inward, the value of 'e' decreases, making the ellipse progressively closer to being a circle. | 677.169 | 1 |
questions and answers, some old some new.
OK, so the original question was about 15 years ago, a WISEA$$ kid sitting in a geometry class (me) says, "this sucks, what are we ever going to use this for?!" Not sure what the answer was then, but can think of something now! This leads to the second question where I ask you guys and hope one of you paid attention. I am not great at explaining stuff and this will be hard to explain, but bear with me. I want to make a table that is in the shape of a capital D if looking down on it. So basically it has 3 sides of a rectangle with the forth side (in place of the second long side) there is an outward arc. I know the dimensions of, the overall length of the table, the width of the 2 smaller sides of the rectangle and I know the width of the widest part of the table which is the center of the arc. My question is how do I find out the radius of the circle that forms the arc? How would I make that? I am imagining some kind of huge compass type of jig that would guide a router to cut the arc? Am I close? Am I way off on this? If so how would you do it, if I am not way off, then how do I figure out that radius? If this isn't clear, you can send me back to the drawing board and I can try to conjure up some images. Thanks a TON in advance and I know, I know, I know the teacher is always right! This is NOT one of the first 500 times I was a JACKA$$ and it came back to bite me! :P :P
Thanks
Joe
RE: questions and answers, some old some new.
Marc is correct in his drawing for an ellipse, however if you want a circle edge, use only the center point.
I just recently saw in one of the WW mags that shows how to construct a jig for your router for cutting a circle or an ellipse. If memory serves me correctly, it invloved two movable arms attached to blocks that moved in a frame in X and Y directions. If anyone remembers the mag (it was just in the past month or so) please post. In the meantime, I'll try to find it tonight and post it later.
RE: questions and answers, some old some new.
Actually, I don't know that I would mess with finding a center for a circle, and given your description, I don't know that a hard fast geometry equation, etc would be in order. By your explanation, the short sides are going to have some straight length before the beginning of the arc forming the rounded side and you know the distance from the center of the long leg to the center of the arc.
.
RE: questions and answers, some old some new.
Joe,
You said that you have three sided of a rectangle and you want the fourth side to be an arc.
Question: What is an alternative approach suggested for finding the center of the circle? Answer: Using the known distance from the center of the long leg to the center of the arc, without necessarily finding a hard fast geometry equation.
Question: What is another method suggested to find the radius of the arc? Answer: Using a jig with two movable arms attached to blocks that move in a frame in X and Y directions, as seen in a recent woodworking magazine.
Question: What are the dimensions Joe knows for the table? Answer: The overall length, the width of the two smaller sides of the rectangle, and the width of the widest part of the table (the center of the arc).
Question: What is the unknown dimension Joe wants to find? Answer: The radius of the circle that forms the arc. | 677.169 | 1 |
No, it is somewhat old and there is nothing online that I can find. – AdamJun 8 '11 at 0:01
If you are doing homework, just use calculator.com or any online calculator. If you need to figure ut how to use the calculator for a test or some other type of assessment, ask your teacher. – JavaManJun 8 '11 at 0:14
It is for a test and for homework, not sure if the teacher has time to answer this question. – AdamJun 8 '11 at 0:15
2
The teacher is paid to have time to answer your questions. – Gerry MyersonJun 8 '11 at 0:26
4 Answers
First make sure your calculator is set to degrees or do the conversion mentioned in the other posts.
You shouldn't need to put in the degrees symbol if the calculator is set to degrees (ie you can see either D or Deg somewhere on the display) if instead you can see R, Rad, G or Grad then you need to set it to degrees instead. If it is set to degrees then the number on which a trig function is operating is already assumed to be in degrees. Your question gives the angle in decimal degrees so you don't need to worry about converting to degrees, minutes and seconds and back again (if you don't understand the last sentence don't worry you probably don't need to).
Older calculators required you to enter the value of the angle 42.0892 first then press the trigonometric button to get the result. Normally if you can see the calculation as you are typing it in then you can just do the calculation in the natural order otherwise you need to do the calculations bit by bit and build up to the final steps.
As a test (with the calculator set to degrees) type 45 and press the tan button, if you have the older style of calculator this will return a value of 1 (the answer to doing tan(45)).
If this is the case then I would work out your calculation using the following steps:
Make sure your calculator is set to degrees
Type in the angle 42.0892
Press tan
The answer is the tan of 42.0892, make it into a reciprocal (there's probably a $\frac{1}{x}$ button or $x^{-1}$ button somewhere on the calculator)
Now multiply the resulting value by the numerator (56.851)
Press equals that should be your answer.
If you can't see a reciprocal button, put the answer to tan(42.0892) into the calculator memory, enter 56.851 and divide it by the stored value.
If you do it right you'll get an answer of 63 rounded to to 2sf (I won't give you the full answer as you should write down the full display and then round it to a suitable degree of accuracy yourself).
Question: What is the final answer, rounded to 2 significant figures, according to the text? Answer: 63 (The text doesn't provide the exact decimal value, but it's implied that the final answer is 63 when rounded to 2 significant figures.)
Question: What is the angle given in the question? Answer: 42.0892 degrees
Question: Is there any online resource mentioned in the text that can help with the calculation? Answer: No, only calculator.com is mentioned, but it's not specified if it's online or not. | 677.169 | 1 |
Lesson Plans & Activities
Line Segments and Angles
This lesson explains lines, points, line segments, rays, and angles. It also details right, acute, and obtuse angles. There are also different student activities for practice including a matching game.
Comments & Collaboration (1)
Select your preferred way to display the comments and click "Save settings" to activate your changes.
This is great!!
I like the interaction you have created with the content. I am wondering if you could give some examples on the pages you have the pull tabs? Some kind of picture... or have the kids draw what they are going to pull, then pull to check. :) Keep up the good work! I can see me using this.
Question: What are the different types of lines discussed in this lesson? Answer: Lines, points, line segments, rays. | 677.169 | 1 |
Conic Sections
Conics or conic sections are the curves corresponding to various plane sections of a right circular cone by cutting that cone in different ways.
Each point lying on these curves satisfies a special condition, which actually leads us towards the mathematical definition of conic sections.
If a point moves in plane in such a way that the ratio of its distance from a fixed point to its perpendicular distance from a fixed straight line, always remains constant, then the locus of that point I called a Conic Section.
The fixed point is called the focus and the fixed line is called directrix of the conic. The constant ratio is called the eccentricity and is denoted by e.
According to the value of there are three types o conic i.e. for e = 1, e < 1 and e > 1 the corresponding conic is called parabola, ellipse and hyperbola respectively.
A conic section or conic is the locus of a point, which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line, not passing through the fixed point.
The fixed point is called the focus.
• The fixed straight line is called the directrix.
• The constant ratio is called the eccentricity and is denoted by e.
• When the eccentricity is unity i.e. e = 1, the conic is called a parabola; when e < 1, the conic is called an ellipse; and when e > 1, the conic is called a hyperbola.
• The straight line passing through the focus and perpendicular to the directrix is called the axis of the parabola.
• The point of intersection of a conic with its axis is called vertex.
• The chord passing through focus and perpendicular to axis is called latus rectum.
• Any chord of the parabola which is perpendicular to the axis is called double ordinate.
• The straight line perpendicular to axis of the parabola passing through vertex is called tangent at the vertex.
Axis of the conic:
The line through focus and perpendicular to the directrix is called the axis of the conic. The intersection point o conic with axis is known as the vertex of the conic.
Enquiry: How do we mathematically define a parabola and what are its various features?
The locus of the point, which moves such that its distance from a fixed point (i.e. focus) is always equal to its distance from a fixed straight line (i.e. directrix), is called parabola.
Equation of Parabola:
Let S be the focus, V be the vertex, ZM be the directrix and x-axis be the axis of parabola. We require therefore the locus of a point P, which moves so that its distance from S, is always equal to PM i.e. its perpendicular distance from ZM. After appropriate configuration let S = (a, 0)
Question: What is the mathematical definition of a parabola? Answer: The locus of a point that moves such that its distance from a fixed point (focus) is always equal to its distance from a fixed straight line (directrix). | 677.169 | 1 |
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing
The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen in the lower diagram part.
If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form two sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore the ratios of their sides must be the same, that is:
This can be rewritten as follows:
This is a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
which is solved to give
And the constant can be deduced from x = 0, y = a to give the equation
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy.
Converse
The converse of the theorem is also true:
For any three positive numbers a, b, and c such that , there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
An alternative statement is:
For any triangle with sides a, b, c, if then the angle between a and b measures 90°.
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
or as follows:
Let ABC be a triangle with side lengths a, b, and c, with Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length c = , the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths a, b and c, the triangles areand must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle.
The above proof of the converse makes use of the Pythagorean Theorem itself. The converse can also be proven without assuming the Pythagorean Theorem.
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and (otherwise there is no triangle according to the triangle inequality
Triangle inequality
Question: How can the converse of the Pythagorean theorem be proven without assuming the theorem itself? Answer: The converse can be proven by constructing a second right triangle with sides of length a and b, and showing that the hypotenuse of this triangle has the same length as the hypotenuse of the original triangle, making the triangles congruent and thus having the same angles.
Question: What is a corollary of the Pythagorean theorem's converse? Answer: A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute by comparing the lengths of its sides.
Question: Why is the constant in the derived equation for the Pythagorean theorem equal to 4? Answer: The constant in the derived equation is equal to 4 because it can be deduced from the initial conditions x = 0, y = a, where a is the length of the side AB in the triangle ABC. | 677.169 | 1 |
Question 745773: Hello, I have this problem where in a parallelogram, one obtuse angle = 225-2x. The x variable represents an acute angle. As you know all the angles of a quadrilateral added up equals 360 degrees. And the angles opposite eachother are equal. I have tried inputting some angles for x such as 45, 35, and 25 but none have worked. Please get back to me ASAP!! I need to know how to figure it out for a test tomorrow! ~Dani Click here to see answer by KMST(1874)
Question 742132: The coordinates of 3 of the vertices of a parallelogram are (–3, 4), (–2, 1), and (2, 6). What is the equation for the line containing the side opposite the side containing the first two vertices? Click here to see answer by KMST(1874)
Question 731845: Ashley wants to use the pattern shown below in her kitchen. The pattern is made up of parallelogram shaped tiles. work out the total area covered by the black tiles. give your answer in cm^2. this is a non calculator question... 2 parallelograms are 200 cm there's a space next to the parallelogram..... then a right angles triangle with a height of 260 cm.
Question 730036: The bases of two parallelograms are the same length. The height of the first parallelogram is half the height of the second.What is the ratio of the area of the first parallelogram to the area of the second? Justify your answer. Click here to see answer by lynnlo(4164)
Question 724170: The measure of an angle of a parallelogram is 18 degrees less than 5 times the measure of an adjacent angle. Explain how to find the measures of all the interior angles of the parallelogram. Click here to see answer by KMST(1874)
Question 723802: Points A, B and C have the coordinates (1,6), (0,0) and (-2,2). Find 3 possible coordinates for a point D so that the four points form a parallelogram . Exclude the case where all points lie in a straight line. How do you work this out? Thanks, Lizzy. Click here to see answer by josgarithmetic(1540)
Question 722681: How to find the area of the shaded region in a triangle if: You know that the triangle is composed of two smaller triangles. And, the first small triangle has the base of 5 and you don't know the height. You just know that the other side of the triangle measures 8. And the second smaller triangle has a base of 4....? Click here to see answer by Edwin McCravy(8909)
Question: What are the coordinates of point D if points A(1,6), B(0,0), and C(-2,2) form a parallelogram? Choose one of the possible answers: A) (3, 4), B) (-3, 2), C) (1, 0). Answer: A) (3, 4) | 677.169 | 1 |
Well it's obvious that this figure IS NOT drawn to the scale, as x=90 degrees and at the figure it's not so.
I asked the similar question to the GMAT tutor Ian Stewart and he kindly gave me explanation. So, below is how GMAT draws the diagrams:
"In general, you should not trust the scale of GMAT diagrams, either in Problem Solving or Data Sufficiency. It used to be true that Problem Solving diagrams were drawn to scale unless mentioned otherwise, but I've seen recent questions where that is clearly not the case. So I'd only trust a diagram I'd drawn myself. ...
Here I'm referring only to the scale of diagrams; the relative lengths of line segments in a triangle, for example. ... You can accept the relative ordering of points and their relative locations as given (if the vertices of a pentagon are labeled ABCDE clockwise around the shape, then you can take it as given that AB, BC, CD, DE and EA are the edges of the pentagon; if a line is labeled with four points in A, B, C, D in sequence, you can take it as given that AC is longer than both AB and BC; if a point C is drawn inside a circle, unless the question tells you otherwise, you can assume that C is actually within the circle; if what appears to be a straight line is labeled with three points A, B, C, you can assume the line is actually straight, and that B is a point on the line -- the GMAT would never include as a trick the possibility that ABC actually form a 179 degree angle that is imperceptible to the eye, to give a few examples).
So don't trust the lengths of lines, but do trust the sequence of points on a line, or the location of points within or outside figures in a drawing. "
Question: Can we trust the scale of GMAT diagrams in Problem Solving questions? Answer: No, we cannot trust the scale of GMAT diagrams in Problem Solving questions. | 677.169 | 1 |
Problem 113.
Area of Triangles, Incircle, Excircles.
Level: High School, SAT Prep, College
In the figure below, given a triangle
ABC, construct the incircle with incenter I and excircles with
excenters E1, E2, and E3. Let be D, E, F, G, H,
and M the tangent
points of triangle ABC with its excircles. If S1, S'1, S2, S'2, S3,
and S'3 are
the areas of the shaded triangles, prove that S1 = S'1,
similarly S2 = S'2, and S3 = S'3.
Question: What are the points D, E, F, G, H, and M in the given triangle ABC? Answer: D, E, F, G, H, and M are the tangent points of triangle ABC with its excircles. | 677.169 | 1 |
You could also say it's a 90+360 = 450 degree angle, or any number $90+360n$ where n is a natural number. The point is, we say that it's 90 degrees clockwise or 270 mostly by convention, but yes, "an angle consists of infinitely many angles" is true.
Similarly for 2) you called it an "equiangular" triangle and so indeed by definition it must have three equal angles, however you measure them.
A "straight angle" would just be an angle of zero (or 360, or 720...) degrees.
Question: What is the relationship between a 90-degree angle and a 270-degree angle? Answer: They are supplementary angles, meaning they add up to 360 degrees. | 677.169 | 1 |
Greatest digit is 9, but less than 900, so at most the number can have three digits, and the hundreds digit cannot be 9. One more than a multiple of 10. All multiples of 10 end in 0, so one more must end in 1. That means that the 1s digit is 1, and since 1 plus 9 is only 10, we need another digit, namely a 2 to make the sum of the digits be 12. The 2 has to go in the 100s place, the 9 in the 10s place, and the 1 in the 1s place.
I am 291.
John
Volume/229296: Please help me with this question...Im not understanding it.--Thank you!!
Two similar triangles are also congruent triangles. 1 solutions Answer 170036 by solver91311(16897) on 2009-10-20 18:04:20 (Show Source):
Perhaps you don't understand the question because it is not a question. It is a statement, an utterly false statement, but a statement nonetheless. Now, if you turn the statement around, you will have a true statement, that is:
Two congruent triangles are also similar triangles.
Your given statement is false because just because the sides of two triangles are in proportion (they are similar), they aren't necessarily equal (congruent). My statement is true because the sides cannot be equal unless they are also in proportion.
John
Volume/229293: I need some help PLEASE.
Two lines are parallel if they have the same slope. Explain why the points with the following coordinates form a parallelogram.
Thank you so much!!! 1 solutions Answer 170034 by solver91311(16897) on 2009-10-20 17:55:57 (Show Source):
The probability of you receiving an answer to this question will increase from 0 to some number larger than 0 if you would only take the time to share the ordered pairs that form the vertices of the quadrilateral in question.
Don't know what an interger is. If you mean integer, then you can't find the greatest integer on a number line. A number line goes on forever, so there is no largest number on it. Of course, you can only represent part of a number line at any given time, so I guess the greatest integer on any given segment of the number line would be the integer with the largest value that you could see on that segment.
Just solve for . However, you will find that there is no solution to this problem because it reduces to the absurdity .
John
Question: What is the last digit of a number that is one more than a multiple of 10? Answer: 1
Question: What is the maximum number of digits a number can have according to the text? Answer: Three | 677.169 | 1 |
Problem 93. Similar Triangles, Circumcircles, Parallelogram.
High School, College
In the figure below, given a triangle
ABC, line DEF parallel to AC and line FGM parallel to AB. If O,
O1, O2, and O3, are the
circumcenters of triangles ABC, DBE, FGE, and MGC respectively,
prove that the quadrilateral OO1O2O3
is a parallelogram.
Question: What is the shape of the quadrilateral OO1O2O3? Answer: It is a parallelogram | 677.169 | 1 |
Given: Triangle ABC is isosceles; Line segment CD is the altitude to the base AB
Could you help me solve this 2-column proof by using statements and reasons please?
Question 211409: Given: angle 1 and angle 2 are supplementary, and angle 3 and angle 4 are supplementary, angle 2 and angle 3 are congruent.
Prove: angle 1 and angle 4 are congruent.
HELP! i'm not sure of the two collum proofs and the Thereoms to go with it. Click here to see answer by RAY100(1637)
Question 210553: Tell whether each statment is true or false. Then write the converse and tell whether it is true or false.
If ,then .
Note: I understand what the converse of this statement is, and I know the answer is false, but I cont comprehend why this converse would be false. Click here to see answer by engchunh(1)
Question 215605: Hi - another question we have is "A line passing through the mid-point of one side of a triangle will passt through the mid point of a second side if and only if it is parallel to the third side of the triangle". We think it has to deal with extending lines to make a quadrilateral and then using congruent triangles...but again we are not sure where to go next...
Thanks Click here to see answer by stanbon(57219)
Question 219819: Moody wants to find the height of the tallest building in his city. He stands 332 feet away from the building. There is a tree 34 feet in front of him, which he knows is 20 feet tall. How tall is the building to the nearest foot? Click here to see answer by checkley77(12569)
Question: Which theorems would be useful to prove that angle 1 and angle 4 are congruent? Answer: The corresponding angles theorem and the supplementary angles theorem. | 677.169 | 1 |
b and another (XA) perpendicular to b.
This process of dividing a
into two parts is known as projectingaonto components parallel and perpendicular tob.
Recall that |a| is the length of OA. The length of OX is therefore . But recall that ,
so that ,
as stated.
Problems
5.9 Let a and b be two vectors. Project a
onto components parallel and perpendicular to b as shown in the picture.
(i) Show that the vector
(ii) Verify that the preceding
result satisfies ,
as it should (why?)
(iii ) Show that the component
of a in a direction perpendicular to
b is
5.10 Let a = 3i+2j+6k, and b = 12i-5j+2k. Calculate the components of a in directions parallel and
perpendicular to b.
5.5 Cross Product(also called the vector product).
Let a and b be two
vectors. By definition, the cross
product of a and b is a vector, denoted by . The direction of c is perpendicular to both a
and b, and is chosen so that (a,b,c) form a right handed triad, as
shown. The magnitude of c is given by
Note that and .
Calculating the magnitude of
the cross product of two vectors is no sweat, but figuring out the direction is
a pain. There are various aide-memoirs
to help you do this- choose the one you find least confusing, or make up your
own.
Right hand rule
To find the direction of ,
arrange your right hand so that your thumb is parallel to a, your index finger is parallel to b, and the angle between your thumb and index finger is . Now set your middle finger is perpendicular
to both a and b. The direction of is parallel to your middle finger. (This rule only really works if ,
otherwise you permanently damage your hand.
Please don't do this.)
Right hand screw rule
To find the direction of ,
arrange your right hand so that your thumb is perpendicular to both a and b, and your fingers curl in the direction of the line joining the
tip of vector a to the tip of vector
b.
The direction of is parallel to your thumb.
Bottle-cap rule.
Obtain a twist-top bottle of
your favorite beverage. Draw an arrow
on the cap. Arrange the bottle so that,
by twisting the cap through the angle ,
you can rotate the arrow from parallel to a
to parallel to b. The direction of is parallel to the direction of motion of the
bottle-cap as it is turned. (Full beverage containers are not be permitted in EN3 examinations)
If none of these tricks help you
Extend your middle finger into
the air. Shout your favorite
expletive. This will not help, but it
may make you feel better.
Problem
5.11 Let {i,j,k} be a
Cartesian basis. Use the definition of
the cross product given above to calculate all possible cross products of the
Question: What is the direction of the cross product of two vectors a and b using the right hand screw rule? Answer: The direction of is parallel to your thumb.
Question: What is the formula for the magnitude of the cross product of two vectors a and b? Answer: The magnitude of the cross product of a and b is given by. | 677.169 | 1 |
Straightedges (guides; curve rulers or templets B43L 13/20; straightedges characterised by the provision of indicia or the like for measuring, e.g. rulers or tapes with measuring scales or marks for direc reading, G01B)
NOTE
-
In this group, the following term is used with the meaning indicated: "straightedge" means an instrument or its edge serving the purpose of acting as a guide for the drawing of a straight line.
Question: What is the primary function of a straightedge according to the text? Answer: An instrument or its edge serving the purpose of acting as a guide for the drawing of a straight line. | 677.169 | 1 |
Question 554048: This is from the geometry regents August 2011, number 38.
Given: Triangle ABC with vertices A(-6,-2), B(2,8), and C(6,-2)
Line AB has midpoint D, Line BC has midpoint E, and Line AC has midpoint F.
Prove: ADEF is a parallelogram.
ADEF is not a rhombus. Click here to see answer by KMST(1936)
Question 554559: help me figure out proofs involving triangles?
Given: m∠3 = m∠7
m∠2 = 60°
m∠6 = 115°
m∠5 = ?
I've figured out that m∠4 = 10, but from there how would i figure out what m∠5 equals? Here's a pic of the triangle below
Question 557341: please help me on this problem
The difference between two supplementary angles is 10. find the acute angle.
i really don't know if this is right but,
s-s=10
(180-x) - (180-x) =10
have no idea if the equation is right, but i know the awnser is supposed to be 85 Click here to see answer by Theo(3504)
Question 561645: I don't know how to approach this proof.
For all numbers a, b,c,d is an element of real numbers.
B,d do not equal zero
if a/b = c/d
then ad=bc
Why mudr the restrictions on b and d exist.
should the same apply to b and d. Click here to see answer by mananth(12270)
Question 562942: How do I proof a parallelogram? I am very confused in proofs. Is there a list I can see on the vocabularies to proof? The question is, Given (square)ABCD
prove: AC and BD bisect each other at E.
Question 563454 Click here to see answer by richard1234(5390)
Question 565708: Two circles meet at points P and Q, and diameters P A and P B
are drawn. Show that the line AB goes through the point Q. (Probably it is easier to
think of drawing the lines AQ and QB and then showing that they are actually the
same line.) Click here to see answer by richard1234(5390)
Question 569917: Write the inverse of the following statement:
If angle A is acute, then its measure is less than 90 degrees
A- If angle A is not acute, then its measure is not less than 90 degrees.
B- If angle A is not acute, then its measure is 90 degrees.
Question: If m∠3 = m∠7 and m∠2 = 60°, what is m∠5?
Answer: 10 degrees (since m∠4 = 10 degrees, and the angles around a point sum to 360 degrees)
Question: In a parallelogram, what is the relationship between opposite angles?
Answer: Opposite angles in a parallelogram are equal.
Question: What are the coordinates of point A in triangle ABC?
Answer: (-6, -2)
Question: What is the sum of the angles in a triangle?
Answer: 180 degrees | 677.169 | 1 |
uses deductive reasoning to justify the relationships between the sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of sides of similar triangles (2.4.A1a).
3.1.A3
understands the concepts of and develops a formal or informal proof through understanding of the difference between a statement verified by proof (theorem) and a statement supported by examples (2.4.A1a).
The student estimates, measures, and uses geometric formulas in a variety of situations.
3.2.A1
solves real-world problems by (2.4.A1a) ($):
3.2.A1A
converting within the customary and the metric systems, e.g., Marti and Ginger are making a huge batch of cookies and so they are multiplying their favorite recipe quite a few times. They find that they need 45 tablespoons of liquid. To the nearest ¼ of a cup, how many cups would be needed?
3.2.A1B
finding the perimeter and the area of circles, squares, rectangles, triangles, parallelograms, and trapezoids, e.g., a track is made up of a rectangle with dimensions 100 meters by 50 meters with semicircles at each end (having a diameter of 50 meters). What is the distance of one lap around the inside lane of the track?
3.2.A1C
finding the volume and the surface area of rectangular solids and cylinders, e.g., a car engine has 6 cylinders. Each cylinder has a height of 8.4 cm and a diameter of 8.8 cm. What is the total volume of the cylinders?
3.2.A1D
using the Pythagorean theorem, e.g., a baseball diamond is a square with 90 feet between each base. What is the approximate distance from home plate to second base?
3.2.A1E
using rates of change, e.g., the equation w = –52 + 1.6t can be used to approximate the wind chill temperatures for a wind speed of 40 mph. Find the wind chill temperature (w) when the actual temperature (t) is 32 degrees. What part of the equation represents the rate of change?
selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, distance, area, surface area, mass, midpoint, and angle measurements (2.4.K1a) ($).
3.2.A2
estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($).
3.2.A3
Question: Does the text mention using deductive reasoning to prove relationships in triangles? Answer: Yes | 677.169 | 1 |
The students will develop basic skills making and identifying homogeneous tessellations, both regular and semiregular.
Materials needed:
One overhead projector, One transparency of tessellation patterns with vertices marked and polygon name listed below, One set of overhead transparency pens, Two - four small plastic bingo chips, One set of plastic regular polygon shapes made from a tessellation pattern consisting of 10 equilateral triangles, 6 squares, 4 octagons, 4 duodecagons, One set of construction paper regular polygon shapes for each student in the class made from the same tessellation pattern as the plastic overhead polygons, One set of 6 to 8 construction paper circles of diameter 1 inch in a color to contrast with the floor of the room being used.
Strategy:
Question: What is the total number of equilateral triangles in the plastic tessellation pattern? Answer: 10 | 677.169 | 1 |
Vertex
A vertex (Latin: whirl, whirlpool; plural vertices), in geometry, is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet).
In 3D computer graphics, a vertex is a point in 3D space with a particular location, usually given in terms of its x, y, and z coordinates. It is one of the fundamental structures in polygonal modelling: two vertices, taken together, can be used to define the endpoints of a line; three vertices can be used to define a planar triangle.
In graph theory, a graph describes a set of connections between nodes. Each node or vertex can map to an object. The connections between the nodes are called edges or arcs
Question: What is the plural form of'vertex'? Answer: Vertices | 677.169 | 1 |
... fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it?
Question: Which mathematical theorem has the author retained from their early studies? Answer: The Pythagorean theorem | 677.169 | 1 |
Try this: start with the vector v0 = (1, 0), and interpret it as the xy-coordinates of a point in the plane. That is, (1, 0) is the point 1 unit to the right of the origin, and 0 units up, so it is on the x-axis. Draw a picture: make a vertical line for the y-axis and a horizontal line for the x-axis, like a big plus sign. Mark off tics every inch or so along the axes, and label them 0 (at the origin where the axes cross), 1, 2, etc. going up (on the y-axis) and right (on the x-axis), and negative numbers -1, -2, etc. going the other direction away from the origin. Now make a dot for the point (1, 0).
Take that matrix A and multiply it by the vector v0 = (1, 0), to get the result v1 = Av0. Make the dot corresponding to that point on your graph. Multiply by A again, and make a new dot for v2 = Av1 = A2v0. Repeat this a few more times, and make dots for v3, v4, and v5.
Now it should make sense to you when you look at the dots on your graph: the matrix A is quite literally transforming the points of the plane in a predictable manner. How would you describe the action of A on vectors?
Now pick some arbitrary vector, like u = (3, 5) but you can pick whatever you like. Mark both the location of your vector u and of Au on your graph. Is Au where you thought it would be?
They're creating radii of circles, like a more confusing version of x^2 + y^2 = 4
This is transformation? Do you always use that specific matrix for this kind of rotation?
I'm starting to get the hang of notation and terminology btw, I think that was half my problem (I just learned what that squiggly equal sign is). Also I also never though of it as some revolutionary unification of geometry and absract algebra, but now that I think about it it's kind of impressive.
There are a family of matrices called "rotation matrices" that happen to rotate every input point by some fixed (for the matrix) angle around the origin.
Calculating them isn't all that tricky -- it involves sin and cos. I think someone posted it earlier in this thread, but deriving it isn't all that hard. If you have a vector (1,0) and you rotate it by 45 degrees, where does it end up? And (0,1), where does it end up? How about a rotation by "sigma" radians?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR
BryanRabbit wrote:They're creating radii of circles, like a more confusing version of x^2 + y^2 = 4 :shock:
Question: How many times is the matrix A multiplied by the vector v0 to get the vectors v2, v3, v4, and v5? Answer: 2, 3, 4, and 5 times respectively.
Question: What is the vector u used as an example? Answer: u = (3, 5) | 677.169 | 1 |
That's basically all of trig, those six functions and their interactions. (You're going to have to memorise those definitions, unfortunately. It helps to notice that adding 'co' to the front simply changes X's to Y's and Y's to X's) And there's a lot of interactions, because X, Y and R are all related to each other by Pythagoras' Theorem: R2=X2+Y2. For example: [sin(theta)]2+[cos(theta)]2 = (Y/R)2+(X/R)2 = (Y2+X2)/R2 = R2/R2 = 1. Thus, [sin(theta)]2+[cos(theta)]2=1, for any theta.
Of course, once you've got the functions, you can throw them into algebra any way you please.
Now, let us say that we wish to rotate a point through an angle of theta. Imagine a line, of length d, before and after rotation. We'll define the "before-rotation" version as zero degrees, and call it B, for Before. I can draw a line, from the end of the "after-rotation" line (A, for After), that hits the "before-rotation" line at right angles. Call this line L, for Line.
Now, L, A and B form a right-angled triangle. In fact, from the point of view of angle theta, and looking at the trig definitions above, A is R, L is Y, and the portion of B between the rotation point and where L hits it is X.
Now, I know how long line A is; it has length d. Thus, R=d. So I look over my trig functions and see which ones will let me find the lengths of B (up to where it hits L) (equal to X) and L (equal to Y). Four of them will do; sin(theta)=Y/R, cos(theta)=X/R, sec(theta)=R/X, and cosec(theta)=R/Y.
I can find Y as either d*sin(theta) or as d/cosec(theta). Either will do. Similarly, I can find X as either d*cos(theta) or d/sec(theta).
Now, using this, can you find the rotation matrix for a rotation through 10 degrees?
Sorry I haven't responded yet ccc, I will get to it (with pencil and paper in hand) because I really want to know trig, right now I'm just obsessed with getting through my algebra book as fast and thoroughly as possible. I see now I really needed a refresher of the basics.
Quick question, when calculating the slope of a line do I always plug in the larger of the two y values for y2? For example if a line passes through points A (0,-2) and B (-5, 12) I know point B is further up on the y axis because it's y-value is higher. That makes it easy because if I make that assumption I know right away m = 12 - (-2) / -5 - 0
Question: Which trigonometric functions can be used to find the lengths of lines B (X) and L (Y) in a rotation scenario? Answer: sin(theta) = Y/R, cos(theta) = X/R, sec(theta) = R/X, and cosec(theta) = R/Y
Question: Which theorem relates X, Y, and R in trigonometry? Answer: Pythagoras' Theorem | 677.169 | 1 |
1. When lines, segments, or raysintersect, they form angles. If the angles formed by two intersecting lines areequal to 90 degrees, the lines are _____________________ lines.
2. Some lines in the same plane do notintersect at all. These lines are _______________ lines.
3. _____________ lines do notintersect, and yet they are also not parallel.
4. ________________ angles are theopposite angles formed by two intersecting lines. These angles have the samemeasure, so they are congruent.
5. A _______________ is a line thatintersects any two or more lines. __________ angles are formed when atransversal intersects two lines. When those two lines are ____________, all ofthe acute angles formed are congruent, and all of the obtuse angles formed arecongruent. These obtuse and acute angles are ______________________.
5-3Terms: Triangles
1. A ________________ triangle has nocongruent sides and no congruent angles.
·algebraic inequality-an inequality that contains at least one variable
d + 3 > 10 5a > b + 3
·solution set-the set of values that make a statement true
·term-the parts of an expression that are added or subtracted
7x + 5 - 3y² + 2x7x, 5, 3y² and 2x are terms
·like terms-two or more terms that have the same variable raised to the same power
7x + 5 - 3y² + 2x7x and 2x are like terms
·simplify-combine all possible operations, including like terms
5a + 7b - 3a + 6a²
5a - 3a + 7b + 6a²
2a + 7b + 6a²
equivalent expressions- expressions that have the same value for all values of the variables; for example when you simplify expressions you are making equivalent expressions during each step of simplification
It is important for your child to reveiw all of the math vocabulary discussed and explored throughout the course of the year. They need to be familiar with the words and meanings in order to solve mathematical problems. They may use their math notebook to brush up on their vocabulary.
Question: What is a triangle with no congruent sides and no congruent angles called? Answer: Scalene triangle | 677.169 | 1 |
Angles can also be measured in radians. At high school level you will only use degrees, but if you decide to take maths at university you will learn about radians.
Figure 2: Angle labelled as B^B^, ∠CBA∠CBA
or ∠ABC∠ABC
Figure 3: Examples of angles. A^=E^A^=E^, even though the lines making up the angles are of different lengths.
Measuring angles
The size of an angle does not depend on the length of the lines that are joined to make up the angle, but depends only on how both the lines are placed as can be seen in Figure 3. This means that the idea of length cannot be used to measure angles. An angle is a rotation around the vertex.
Using a Protractor
A protractor is a simple tool that is used to measure angles. A picture of a protractor is shown in Figure 4.
Figure 4: Diagram of a protractor.
Method:
Using a protractor
Place the bottom line of the protractor along one line of the angle so that the other line of the angle points at the degree markings.
Move the protractor along the line so that the centre point on the protractor is at the vertex of the two lines that make up the angle.
Follow the second line until it meets the marking on the protractor and read off the angle. Make sure you start measuring at 0∘∘.
Measuring Angles : Use a protractor to measure the following angles:
Figure 5
Special Angles
What is the smallest angle that can be drawn? The figure below shows two lines (CACA and ABAB) making an angle at a common vertex AA. If line CACA is rotated around the common vertex AA, down towards line ABAB, then the smallest angle that can be drawn occurs when the two lines are pointing in the same direction. This gives an angle of 0∘∘. This is shown in Figure 6
Figure 6
If line CACA is now swung upwards, any other angle can be obtained. If line CACA and line ABAB point in opposite directions (the third case in Figure 6) then this forms an angle of 180∘∘.
Tip:
If three points AA, BB and CC lie on a straight line, then the angle between them is 180∘∘. Conversely, if the angle between three points is 180∘∘, then the points lie on a straight line.
An angle of 90∘∘ is called a right angle. A right angle is half the size of the angle made by a straight line (180∘∘). We say CACA is perpendicular to ABAB or CA⊥ABCA⊥AB
. An angle twice the size of a straight line is 360∘∘. An angle measuring 360∘∘ looks identical to an angle of 0∘∘, except for the labelling. We call this a revolution.
Question: Which of the following is NOT a way to measure angles, according to the text? A) Using a ruler B) Using a protractor C) By rotation around the vertex Answer: A) Using a ruler | 677.169 | 1 |
Angles/457940: two angles are supplementary. one is 87 degrees more than twice the other. Find the measures of the angles. 1 solutions Answer 314115 by rfer(12657) on 2011-06-06 00:10:41 (Show Source):
You can put this solution on YOUR website! you missed the -5
and didn't move the three correctly.
Your having a little trouble with signs.
12x+8-x-5=-3
11x+3=-3
11x=-6
x=-6/11
Good luck
Bob
Money_Word_Problems/457766: A salesman earns $250 per week PLUS a 10% commission on all sales over $5000. In one week, his sales totaled $14,000. What were his earnings that week? 1 solutions Answer 314034 by rfer(12657) on 2011-06-05 16:51:25 (Show Source):
Question: What is the difference in degrees between the two angles? Answer: 87 degrees | 677.169 | 1 |
So all this is just so that depth can be taken into account? Speaking of which your diagram can either be considered wrong or your suggestion lacking details. There is no measurement of the distance away from the monitor. Especially when for people like me, having the hydra base close to the monitor causes jitter. Which is strange cause I have an LCD, but oh well.
Since the diagram becomes useless without the monitor, I'm gonna go with missing details. The alternative means there is no right angle in the triangle (B = base; H = hand; Y = you)
If you don't mind doing it with non right angles have at'r
I don't mind proving a trig identity, but calculating stuff.. not so much. In any case the non right angle triangles are more accurate to how things actually are.
Edit: The more you open the angle (ie. bigger angle) the slower it should move actually. You place your hand closer to the monitor in your diagram and the angle becomes wider. Then it becomes a much larger motion of your wrist to point to either side of the monitor.
Question: Is the diagram provided in the text accurate for measuring depth? Answer: No, the diagram is not accurate for measuring depth as it lacks the measurement of the distance away from the monitor. | 677.169 | 1 |
Hey there. I am not sure if my other comment, but here's what I wanted to say. My example in Math 1060 for angles was the "we use angles to measure the distance from the where I am standing on the horizon to where the moon is in the sky." Well in my astronomy book, all I could find was angular distance which they use that to measure how wide an object is, or to measure the distance between stars. Is that what you were thinking too? Also a little measuring fun, to measure the angular distance of any object in the sky, you use one of your hands: have your hand facing away from you, fingers fanned out, 1 degree is your the width of your index finger, 10 degrees is the span of you hand from below your pinky to below your index finger (thumb not included). I can show you in class 3, 4 and 5 degrees sometime. Maybe you could show Griffy someday because of his love of measurements. But back to the question, is "angular distance" the term we want to use for what I was talking about?? Ang
Question: Who does the user suggest showing the hand measurement method to? Answer: Griffy, due to his love of measurements. | 677.169 | 1 |
timeline should contain all the required information listed in Section A of the project.
Section A: Timeline
Please note that some the times given are approximations, historians provide slightly different dates for the earlier developments in trig. There is also disagreement about which trig table should be counted as the first. I have included some of the Chinese developments in trig – this happened independently to the other developments – it is not essential that learners include this info.
Table 1
Approx. 1600BCE
Egyptians & Babylonians: Used ratios of sides of similar triangles, known as trilaterometry
Egyptian: Ahmes (1680-1620) – Trig is used in Pyramids
Babylonian: Tablet called Plimpton 322 (dated 1800-1600BCE) gives a table of values. It is argued that it could be a table of secants, however some say it is more likely a table of Pythagorean triples. or - which shows that they Pythagoras' theorem was in use before his time (530BCE)
Babylonians introduced the idea of angle measurement
200BCE
Greek: Started making use of chords, and noticed ratios/patterns associated
140BCE
Greek: Hipparchus of Nicea – wrote 12 books on chords of a circle. He developed the first trigonometric table. This is seen as the first work on trig. It has since been lost, however historians agree that it did exist. Some refer to him as the 'father of trigonometry'.
Greek: As a result of Hipparchus' tables mathematicians started to work with 360 degrees in a circle (the original idea of breaking a circle into 360 parts is attributed to Hypsicles, a Babylonian Astronomer: 190-120BCE).
100
Greek: Mathematician Ptolemy writes The Almagest, which contains tables of chords and includes explanation on how develop tables and gives examples of solving triangles using chords. He also develops the half-angle theorem.
Indian: Idea of sin and cos starts to develop (as opposed to use of chords)
C5
Indian: Mathematician Aryabhata (476–550 AD), defined the sine as the relationship between half an angle and half a chord. Also defined cosine, versine, and inverse sine. He developed tables for sine and 'versine' values that are still around today. He used the words jya for sine, kojya for cosine. (The words jya and kojya eventually became sine and cosine respectively after a mistranslation.)
C6
Indian: Varamihira expands Aryabhata's work on trigDevelops various formulae
C7
Indian: Brahmagupta's formulas expands Varamihira's work to include other formulae
Europe: Islamic works are translated and Europeans begin to build on previous work. Regomontanus: treated trigonometry as a distinct discipline
Question: Which ancient civilization used ratios of sides of similar triangles? Answer: Egyptians and Babylonians
Question: What did Aryabhata define as sine and cosine? Answer: He defined sine as the relationship between half an angle and half a chord, and cosine as the adjacent side of a right-angled triangle.
Question: Who was the first to treat trigonometry as a distinct discipline? Answer: Regomontanus | 677.169 | 1 |
Trisecting the Area of a Triangle
by
James W. Wilson
Introduction.
The task is to divide a given triangle into three regions of equal
area, using line segments and points. There are several different
problems that can be posed.
Here are four:
Problem 1. If a triangle ABC is given and a random
point P on the triangle is selected, construct two lines through
P to divide the triangle into three regions with equal area. At the right is ONE example but because P can be anywhere on the perimeter of the triangle there are several cases to consider. We will enumerate 7 of them below.
Problem 2. Given a triangle ABC, find a point D such that
line segments AD, BD, and CD trisect the area of the triangle
into three regions with equal areas. Define D and prove that
the triangle is divided into three regions of equal area. Show
a construction for finding D.
Problem 3. Given a triangle ABC, and given a
point E. such that line segments AE, BE, and CE trisect the area
of the triangle into three regions with equal areas. Show a construction
and prove that it divides the triangle into three regions of
equal area.
Problem 4. Given a triangle ABC. Construct two line
segments parallel to the base BC to divide the triangle into
three regions with equal areas. Prove that the construction divides
the triangle into three regions of equal area.
Problem 1.
If there is given a point P on the side of the triangle, then
there are seven ways to construct three equal areas with lines
drawn from P. The sequence below shows the seven patterns with
P moving from right to left.
In the first case, P is the base vertex on the right hand side
of the triangle, and in the seventh case, P is the base vertex
on the left hand side of the triangle. The construction in each
case involves trisecting the segment on the opposite side of the
triangle and connecting the vertex by segments to the two trisection
points. Since we know at least five ways to construct the trisection
of a line, the construction task for these two cases can be assumed.
Much needs to be discovered in the other cases.
For the third and the fifth cases, P is located such that a segment
drawn to the opposite vertex determines a triangle that is one-third
the area of the original triangle. This means P is located at
a trisection point on the base of the triangle. But then the remaining
two-thirds of the original triangle must be divided into two equal
areas by a line from P through the opposite side, making two triangles.
These triangles have the same height, so the point must be the
midpoint of that side. That is, if P is a trisection point of
the base, then the original triangle is divided into three equal
areas by lines from P to the vertex and from P to the midpoint
of the side along the two-thirds section.
This leaves three constructions to be determined, the second,
Question: What is the topic of the text? Answer: Trisecting the Area of a Triangle
Question: What is the location of point P in the first case of Problem 1? Answer: P is the base vertex on the right hand side of the triangle. | 677.169 | 1 |
Triangle BAD is congruent to Triangle CAD (statement) Given (reason)
AD is perpendicular to BC (statement) Given (reason)
BA is congruent to AC (statement) Corresponding parts of congruent triangles (reason)
You can put this solution on YOUR website!
If we assume that the labeling in the problem ("Given triangle BAD is congruent to triangle CAD") is correct, then right away we can claim BA = CA and triangle ABC is isosceles (you might want to say BA = CA --> ABC is isosceles in your proof).
If the labeling is incorrect (e.g. triangle BAD is actually congruent to triangle CDA") then that is a mistake on the textbook. Just make sure you also label your points correctly, matching corresponding ones next to each other.
Either way, you would want to specify that BA = CA implies that triangle ABC is isosceles.
decimal-numbers/488014: the surface area of the moon is 37,900,000 square kilometers, which has larger surface area?
You can put this solution on YOUR website! The question "Which has larger surface area?" implies you have to choose between two or more objects or things. You only posted one "thing," the surface area of the moon. We need another object to compare it to. Post your entire question next time.
While these ordered pairs do not have a unique solution, we can find a continuous function that models this. The function appears to be symmetric about x = 0, and by noticing a pattern we can tell that y = x^2 + 1 appears to be a solution.
test/488093 1 solutions Answer 333282 by richard1234(5390) on 2011-08-30 00:33:41 (Show Source):
Question: If the labeling in the problem is correct, what can we claim about BA and CA? Answer: We can claim that BA = CA. | 677.169 | 1 |
The
horizontal lines of latitude are called parallels
because they run parallel to the equator. Imagine them as
horizontal "hula hoops" around the earth.
The
latitude line numbers measure how far north or south of the
equator a place is. The equator has the number 0 degrees latitude.
The numbers get larger the further away from the equator up
to 90 degrees. Latitude locations are given as __ degrees
north or __ degrees south.
Question: What is the range of latitude numbers? Answer: 0 to 90 degrees | 677.169 | 1 |
Ellipse
In the following figure the plane
is slicing the cone at an angle,β,
greater than 900.
As β increases the plane will eventually become
parallel with the edge of the cone. At this point the section will be a
parabola; up to this point we have an ellipse.
The
intersection will look like the following:
An
ellipse has two axes, the major and the minor axes. Each half is
called a semi axis and are the radii of the ellipse.
There are two focal points,
always on the major axis, at the same distance from the center of the
ellipse. This distance we'll label with the lower case letter
c. The significance of these focal points will be discussed later.
The length of the x radius we'll label with
the letter a. Now, imagine sliding the
ends of the dark red line along both axis through points 1, 2 then 3.
The length of the y radius we'll label
with the letter b (shown below.)
The distance from f1 to P is also a. The triangle f1 p f2
is an isosceles triangle with a base equal to 2c
and whose legs each equal a and whose
altitude is b.
The distance from f1
to any point P on the ellipse to f2 is constant, equal to
2a. Of course, this is true for every ellipse.
The major axis can lie along the
Y axis as shown below. All previous relationships hold.
In each ellipse above (horizontal and vertical) a, b, and c form the sides of a right triangle. a
and b interchange rolls as the hypotenuse. Using Pythagorean's
Theorem, we have
b2 + c2 = a2
and a2 + c2 = b2
c2 = a2 - b2
and c2 = b2 - a2
c =
(a2 - b2)½and c =
(b2 - a2)½
We can combine these two equations using absolute
value like so:
c = | a2 - b2 |
½
When you look at the horizontal and vertical ellipses you'll note that in
the first a is always larger than b, and in the second b is always
larger then a. We're interested in the difference of their squares,
and that must be positive, so in both cases the absolute value of this
difference will work.
As with the circle, we can translate the center of the ellipse to a point (h,k)
and the like this:
and the equation in this case is :
CAUTION:
most, if not all texts will take this standard equation and interchange the a
and b to keep the 'a' with the major axis and the 'b' with the minor axis to
describe a vertical ellipse.
Like so:
I believe that doing this only adds
confusion. I chose NOT to identify a and b with these axes but, instead, with the
Question: What is the standard equation of an ellipse with its center translated to a point (h,k)? Answer: (x - h)²/a² + (y - k)²/b² = 1 | 677.169 | 1 |
I have two intersecting quadrilaterals (the area of intersection is the grey polygon with thick boundary):
These properties holds:
One quadrilateral is always a rectangle
There is always some intersection
Both quadrilaterals are convex (hence the intersection is a convex polygon as well)
The goal is to measure area of intersection (the actual shape is not needed, only a scalar showing how much space is covered by the intersection).
The problem arises in computer graphics (image mosaicing using projective geometry), where one image is stationary and the other is rotated in space and then projected in the same plane as the first one. I need to sort the images according the area of intersection they form.
You could omit step 4 and compute the area using the shoelace formula as a cyclic sum of $2\times2$ determinants $x_iy_{i+1} - x_{i+1}y_i$. YOu still need to order the triangles to form a convex polygon, though. – MvGJul 9 '12 at 23:01
Question: In which field does the problem of measuring the area of intersection arise? Answer: Computer graphics, specifically in image mosaicing using projective geometry. | 677.169 | 1 |
Social Web Research
This page contains a list of user images about Phase Angle which are relevant to the point and besides images, you can also use the tabs in the bottom to browse Phase Angle news, videos, wiki information, tweets, documents and weblinks.
In the context of vectors and phasors, the term phase angle refers to the angular component of the polar coordinate representation. The notation for a vector with magnitude (or amplitude) A and phase angle θ, is called angle notation.
In the context of periodic phenomena, such as a wave, phase angle is synonymous with phase.
Question: What is the phase angle called in the context of periodic phenomena, such as a wave? Answer: Phase | 677.169 | 1 |
Once you start thinking about ways to give students more power/responsibility, you see them everywhere. What have you done to change the balance of power in your classroom – either towards the students or towards the teacher?My blog has gotten a little lofty lately, and it's been a while since I just posted some plain old good ideas you could use tomorrow. Here's one if you have access to a class set of laptops or a computer lab: have your students make tools in geogebra. I'm not going to try to frame this in a lesson plan – it's just a tutorial for you. Open geogebra to follow along.
We'll make a midpoint tool1 in two different ways today. The first way will be geometrically, via construction. The second way will use an algebraic formula. This might be a fun way to connect geometry and algebra! If you already know both of these methods, skip down to the "Toolify" section. If you already know how to make tools, skip down to "The Point" section at the bottom!
Midpoint via Construction
To make tools in geogebra, you first do what you want the tool to do, and then tell geogebra about it. So to make our midpoint tool via construction, we first have to actually do the construction.
Start with two points.
Use the circle tool to draw a circle with center A and perimeter point B. Draw a second one with center B and perimeter point A.
The circles intersect at two points. Use the line tool to draw the line between them. Also draw the line from A to B.
Of course, these two lines intersect at the midpoint between A and B. Use the point tool to give it its own name.
A crucial step: test your construction by moving the points A and B. The entire construction should move, but E should still be the midpoint. Do not move on if your construction does not pass this "wiggle test."When your construction passes the wiggle test, go to the "Toolifying" section below.
Midpoint via Algebra
We'll do this construction entirely from the input bar. Text in bold is text you can type directly into the input bar.
A = (2, 4)
B = (5, 6) Typing these commands creates two points, A and B, at the specified coordinates.
x_A = x(A)
y_A = y(A)
x_B = x(B)
y_B = y(B) These commands create variables with which you can access the coordinates of points A and B. The thing on the left of the equal sign is the NAME of the variable. The thing on the right of the equal sign is the VALUE of the variable.
Do the wiggle test on your variables. When you wiggle points A and B, all four of the variables from step 2 should change. You can move point A by dragging it with the mouse, or by redefining it with something like A = (1, 4). Do not move on until your variables have passed the wiggle test.
Question: What is the next step after passing the wiggle test in the construction method? Answer: Proceed to the "Toolifying" section
Question: What is the purpose of the wiggle test in the algebraic midpoint method? Answer: To ensure that the variables change when points A and B are moved.
Question: Which command creates a variable to access the x-coordinate of point A? Answer: x_A = x(A) | 677.169 | 1 |
We'll each eat through the Radius,
To the center where we get a Lady and Tramp kiss.
The distance from one side, through the middle to the other side
Is the Diameter, so never say die. Perimeter, distance around a circle is 2 pi R. Area is pi R squared,
How much space is in there, do you care?
Well I sure hope you do,
Or I'll be eating pancakes, and you'll be eating Froot-Loops.
It's all mathematics. We add like addicts.
We subtract like taxes, we multiply like rabbits.
We divide like axes. It's all mathematics
polygon - a closed shape with straight sides
regular polygon - a polygon where all sides and all angles are equal
triangle - a shape or polygon with three sides
quadrilateral - a shape or polygon with four sides
pentagon - a shape or polygon with five sides
hexagon - a shape or polygon with six sides
heptagon - shape or polygon with seven sides
octagon - a shape or polygon with eight sides
nonagon - a shape or polygon with nine sides
decagon - a shape or polygon with ten sides
decimal - a fraction written with a "decimal point," like .
5 rather than 1/2
decimeter - one tenth of a meter
congruent - equal, generally relating to two geometric figures that are identical in size or shape
equidistant - equally distant from two points
adjacent - directly next to
perpendicular - two lines that meet at a 90 degree angle
parallel - two lines that will never cross
plane - a two dimensional shape that extends infinitely in all directions
circle - a round shape where all points are equidistant from the center
radius - the distance from the center of a cirle to its circumference
diameter - a line that passes through the center of a circle and cuts it in half
perimeter - the distance around the outside of a shape. The perimeter, or circumference, of a circle =2 pi r
area - the amount of space within a two dimensional shape. The area of a circle = pir R squared.
What is the word for any shape with a bunch of sides?
Polygon
What is special about a regular polygon?
All sides are the same length.
How many sides does a heptagon have?
Seven
An octopus has the same number of legs as what shape?
Octagon
What does "congruent" mean?
Equal
When two lines will never cross, they're _____.
Parallel
What dimension is a plane?
2-D
In what shape is every point equidistant to the center?
A circle
The length from the center of a circle to one end is called the _____.
Radius
What do we call the distance around the circle?
Perimeter
Question: What does it mean for two lines to be perpendicular? Answer: They meet at a 90 degree angle.
Question: Which of the following is NOT a polygon? A) Triangle B) Circle C) Quadrilateral D) Pentagon Answer: B) Circle | 677.169 | 1 |
Question 212281: I have to figure out the measure of the angles. The problem is "find the measure of angle T if the measure of angle T is 20 more than four times its supplement" I know the answer is 148 but I can't figure out how to get to that. I tried solving it by setting up the equation 4t+20=180 but I got 40. And I just really can't figure out how to so this. Please help!! Click here to see answer by jim_thompson5910(28476)
Question 212339: This is the problem out of the book. "Angle L and Angle M are complementary angles. Angle N and Angle P are complementary angles. If the measure of angle L=y-2, the measure of angle M=2x+3, the measure of angle N=2x-y, and the measure of angle P=x-1, find the values of x, y, and the measures of angles L, M, N, and O." Please help me I don't even know where to start. Click here to see answer by [email protected](15624)
Question 212933: a man has a 30 degree, a 60 degree, a 150 degree, a 45 degree and a 135 degree angle in his pocket. he removes two of the five angles from his pocket. find the probability that:
a) the two angles are supplementary
b) the two angles are complementary Click here to see answer by Theo(3458)
Question 212955: The question given to me was " find the measure of an angle whose supplement measures 2 degrees more than three times the measure of it's complement."
I understand what complimentary and supplementary angles are, but I am confused on how to complete this problem. Please don't just give me the answer, I need to be able to explain how to do these types of problems. Click here to see answer by Earlsdon(6287)
Question 212961: I have a picture with segment XY is perpendicular to segment YW. Ray YZ forms two angles within the 90 degrees of angle XYW. these ar identified as angle x (angle XYZ) and angle y (angle ZYW). The next picture i have has segment AB is perpendicular to segment BC. A segment BD is within this 90 degrees showing angles (2x)degrees (angle ABD) and (y-50) degrees (angle DBC). I need to find the measure of angle DBC.
Question: What is the measure of angle DBC in the fifth problem? Answer: (y-50) degrees | 677.169 | 1 |
A faster way to do it would be to use the equation for regular polygons:
180(n-2)=total # of angles
where n is the number of sides
And the total number of angles is equal to the number of sides multiplied by the angle of each side (call x the measure of each interior angle, and so it would be xn)
From x + y = 80, you can then figure out that each angle is 140 degrees.
Then just plug into the equation
180(n-2)=140n
180n-360=140n
40n=360
n=9
So there are 9 sides.
Question: What does 'n' represent in the formula? Answer: The number of sides of the polygon | 677.169 | 1 |
Napoleon's Theorem Geometry
WhiteBrownPaquitaFiona asked
I have a question on the proof. This is the beginning of the proof.
The theorem states that if you have a triangle ABC and you construct equilateral triangles on each of the three sides, then the three centers of those equilateral triangles always form an equilateral triangle themselves.
The question is: When we inscribed the equilateral triangles in a circle, why do the three circles intersect at one point? Can you please help me understand more the proof? I'd highly appreciate an adequate response! Thank you!
Question: What does the user express at the end of their message? Answer: Gratitude (they say "thank you") | 677.169 | 1 |
Point-in-polygon: Jordan Curve Theorem
Calculating whenever a point is inside a polygon can sometimes be a hard and costly calculation. This article describes a quite cheap solution to calculate whenever a point is inside ANY closed polygon. In an open polygon it's hard to determine what's in and out so naturally it won't work.
A closed polygon with 3 points marked
The Jordan Curve Theorem states that a point is inside a polygon if the number of crossings from an arbitrary direction is odd. An image explains more than a thousand words so lets take a look at the picture. As you can see point 1 and 3 is inside the polygon but point 2 isn't. Follow the rays from each point and count each time you cross a line-segment. In this article I only deal with 2D polygons but it can easily used in a 3D-environment.
Contents
One of the first things to do is to cast a ray from the point in an arbitrary direction. I use a ray along the Y axis (pointing upwards as in the picture) for simplicity. Along X-axis is good too, but use one of those or it gets a lot harder. Remember that I use the ray mentioned above throughout this article.
As a first step, if the ray is along y-axis, check if the point x-coordinate is between the two points connecting the line. If not it don't cross it either. You can also check if the y-coordinate is above both points. The next step is to find the equation of the line. Hopefully you remember this from grade school. The equation of a straight line is (Swedish notation). The slope is and offset is . Do the math we have the equation. Now, insert the x-coordinate of the point into the equation. If the result is larger than the y-coordinate the ray does not cross the line-segment.
Question: What is the equation of a straight line in the given text? Answer: The equation of a straight line is y = mx + b, where m is the slope and b is the offset. | 677.169 | 1 |
Lets see how: First draw a perpendicular from the x-axis to the point P. Lets call the point on the x-axis N. Now we have a right angle triangle \triangle NOP. We are given the co-ordinates of P as (-\sqrt{3}, 1). i.e. ON=\sqrt{3} and PN=1. Also we know angle PNO=90\textdegree. If you notice this is a 30\textdegree-60\textdegree-90\textdegreetriangle with sides 1-sqrt3-2 and angle NOP=30\textdegree.
The coordinates of point P is (-\sqrt{3}, 1) ie ON = 1 and PN = root3 - not the other way round as explained in his posting quoted in blue above.I understood bunuel's solution ok but i have a problem with the other one.
I got every step of maths but how did you get (t-1) too. However How did you decide that "t" is bigger than 1. Because it can be less then one then it should became (1-t). Can anyone explain it plz just that part??
OP and OR are prependicular to each other. Hence slope of one must be negative inverse of the other. Slope of OP = -1/sqrt{3} Slope of OQ = s/r and it should be equal to sqrt{3}. Hence s = r * sqrt{3} -----------------(1)
Since OP and OQ represent radius of the circle, their lengths must be equal. Length of OP = sqrt{(sqrt{3})^2 + 1^2} Length of OQ = sqrt{r^2 + s^2}
OP = OQ, Also substitue the value of s obtained in (1). Upon solving, we'll get two values of r +1 and -1. Now since r lies in quad I r has to be +ve. Hence +1.
In the figure above, points P and Q lie on the circle with center O. What is the value of s?
This is how I solved it.. This will take less than a minute...
Drop perpendiculars from P and Q. Mark the points as X and Y.
Now, PXO and QYO are right angled triangles.
We know, PO =1 and XO = sqrt(3). So PO = radius = 2.
XOP + POQ + QOY =180 so, QOY = 30, This gives, t= sqrt (3) and s = 1.
Yeah this is the quickest way I think... But I think the underlined portion is stated wrongly....
The method with the angles is the quickest one. However, I solved it using the slopes. PO and QO have negative reciprocal slopes. The slope of PO=\frac{-1}{\sqrt{3}} The slope QO is the native reciprocal to PO:\sqrt{3} This means that we have to solve the system: \frac{t}{s}=\sqrt{3} r^2+t^2=4
Question: What are the possible values for r? Answer: +1 and -1
Question: What is the length of PN? Answer: 1 | 677.169 | 1 |
That is, if the boat heads directly across, its path makes an angle of 51 degs with the line perpendicular to the shore.
Probability-and-statistics/445561: In a scale drawing of a house, the living room is 1 in. long. The actual length is 24 ft.
On the drawing, a hallway is 1/4 in. wide. Find the actual width.
Question: What is the angle that the boat's path makes with the line perpendicular to the shore? Answer: 51 degrees | 677.169 | 1 |
Reciprocal Runway Math Made Easy
Reciprocal Runways – 180 Degree Opposites
Runway Numbering
The diagram shown here displays a 4,000 foot long (75 foot wide) runway with two ends. Each end of the runway is labelled with a large number. The actual pavement would have these large numbers painted on the runway at the threshold for each end of the runway. In this example, we see the Runway Numbers 3 and 21.
Magnetic Direction
The runway number represent the first two digits of the runway's actual three digit magnetic direction. Runways are oriented or pointed at angles with respect to the magnetic north pole. The magnetic north pole is where your compass points, and not the true north pole. The Runway Numbers are essentially the runway direction, rounded off to the nearest 10 degrees.
Reciprocal Runways
By now, you have probably figured out that the Runway Diagram shown here identifies two runways sharing the same pavement. There are two ends to the pavement, and each end is a Reciprocal Runway of its opposite end of the pavement. For instance, Runway 3 and Runway 21 on the diagram point in opposite directions. Runway 3 is at a 30 degree angle in relationship to the magnetic north pole, and Runway 21 is at a 210 degree angle. The difference between these two angles is 180 degrees. 30 + 180 = 210.
180 Degrees
When you are on a Runway, you can calculate the opposite direction (Reciprocal Runway) by adding or subtracting 180 degrees. Every circle has 360 degrees, and therefore 180 degrees is exactly half of the full rotation of a circle. As you see, Runway 3 (30 degrees) is paired with Runway 21 (210 degrees), and these two runways are 180 degree opposites.
Runway 28
Here we see Runway 28. As only the first two digits of the three digit magnetic direction are used for Runway Numbering, we realize this Runway is aligned at 280 degrees from Magnetic North.
Reciprocal of Runway 28?
You could calculate the Reciprocal of Runway 28 (280 degrees) by subtracting 180 degrees. This would be 100 degrees, or Runway 10. If a Runway is between 0 and 180 degrees, then add 180 degrees to calculate the Reciprocal runway. However, if the runway is above 180 degrees, simply subtract 180 degrees to calculate the reciprocal runway angle.
Math Made Easy!
Here's a little trick…
Question: What is the reciprocal runway of Runway 28? Answer: Runway 10
Question: If a runway is above 180 degrees, how do you calculate its reciprocal? Answer: Subtract 180 degrees | 677.169 | 1 |
You could start with this instead, as sort of a magic trick. "I can make the bull's-eye stay in focus when I toss this shape across the room, or I can make it blurry." Amaze them with your ability to make it happen the opposite way from what they predict every time, until they figure out that you're showing them different sides. (This works best if the "wrong" bull's-eye is not tremendously far from the centroid, of course, so the students hopefully won't immediately catch that the two sides are different. Now, I don't know how that leads your students to figuring out that the center point of the side that works is the intersection of the medians, so this method may not be useful for your purposes.
Similar to Alex's approach, I'd start off with an already drawn triangle and ask them how to exactly double that triangle's area by adding on another adjacent triangle. (You will need to work on the phrasing a bit.) Essentially, get them to do the task backwards, and then give them a new triangle (acute, obtuse, or right) and ask them now to find out how to divide up the area equally.
Put a point on the base of the triangle that isn't the midpoint, then draw the two triangles formed. Ask "Which of these two triangles has more area?"
Then move the point and do it again. Ideally put it somewhere that looks suspicious. If you want, you could use an angle bisector -- my personal favorite is the angle bisector to the "24" side of a 7-24-25 triangle, which is 5.25 units along the 24 side. (Sorry, this needs a picture, and sorry that I have a favorite angle bisector.)
Do it one more time, this time with a new triangle where you label only the base and not the height, hoping kids may notice that it only depends on the bases chosen.
Then ask "where can we put the point so that both areas are equal?" and it should work.
Question: What is the initial "magic trick" described in the text? Answer: The trick involves tossing a shape with different sides across a room, making a bull's-eye appear in focus or blurry, contrary to what observers predict.
Question: What is the expected outcome when students answer this final question correctly? Answer: They should be able to divide the area of the triangle equally. | 677.169 | 1 |
Question 617271: A cruise ship sailed north for 50 km. Then, the captain turned the ship eastward through an angle of pi/6 and sailed an additional 50 km. Without using a calculator, determine an exact expression for the direct distance from the start to the finish of the cruise. Click here to see answer by nerdybill(6948)
Question 617270: The wings on an airplane are usually angled upward relative to the body, fuselage, of the aircraft. This is known as dihedral, and makes the aircraft more stable in flight, especially during turbulence. Some aircraft hve double dihedral. Suppose that wing is designed as shown, with an initial dihedral angle of x for the first 6ft and an additional angle of x for the next 6 feet. Show that h2= 6sinx(1+2cosx).
... I have no idea where to start with this question. Click here to see answer by KMST(1852)
Question 617824: If sin θ = 0.42 then what does sin (180+θ) equal?
I recently did a trigonometry test and this was one of the questions on the non-calculator section. I know how to work it out using a calculator but I'm not sure how to do it without one. Click here to see answer by lwsshak3(6456)
Question 619105: please help:
the height of a radio transmission tower is 70 meters and casts a shadow of 30 meters. draw a diagram & find the angle of elevation of the sun?
i understand u may not be able to draw the diagram. thanks:) Click here to see answer by scott8148(6628)
Question 6 Click here to see answer by mathgranny(7)
Question: What is the total distance traveled by the cruise ship if we consider only the horizontal components of its movement?
Answer: 100 km (50 km north + 50 km east)
Question: If sin θ = 0.42, what is the value of cos θ?
Answer: 0.78 (using the identity cos^2 θ + sin^2 θ = 1) | 677.169 | 1 |
In this problem, you are given a set of points. By combining these points in given order by lines you get a shape. You need to check whether it matches with S shape which could be rotated by 90, 180, 270 degree.
Depending on the type of shape it matches you need to return that number. For example if it matches with shape 1 of the diagram return 1. If it doesn't matches any return 0.
The prototype of the function is:
int matchshapes(struct point p[],int size); where p is given array of points, size represents array size
Constraints
A given set of points that makes S shape has its all lines that are parallel either x or y axis.
Example 1
Question: What should the function return if the shape matches with a rotated 'S' shape? Answer: The function should return the number corresponding to the matched rotated 'S' shape (1, 2, 3, or 4). | 677.169 | 1 |
join the community, vote, and change the world.
If you hit "shift" while drawing a rectangle in Microsoft Paint, it will always make a perfect square. Similarly, holding "shift" while drawing an ellipse will make a perfect circle, and holding it while drawing a line will cause the line to be perfectly horizontal, vertical, or at a 45o angle.
Question: Which of the following shapes can you make perfect by holding the "shift" key while drawing in Microsoft Paint? A) Triangle B) Rectangle C) Ellipse D) All of the above Answer: D) All of the above | 677.169 | 1 |
Special Functions
Trigonometric Functions
Graphing Trigonometric Functions Page [1 of 3]
Now, what about the graphs of these functions. How would you actually graph them? Because we'll be looking at graphs of these trigonometric functions, so where do we go from here? Well, the graphs actually - suppose we wanted to graph, let's say the sine function. So F of X equals sine of X, but before we even graph that, I've got to tell you about the x's here.
Traditionally we think about angles measured in degrees, and that's great, and as you remember, one cycle around represents 360 degrees. And we feel very comfortable with that, we know that this right angle is 90 degrees. If you just go one over, and do a straight line that's 180 degrees. But why should once around be called 360 degrees? Sort of an interesting question to think about, right? Why 360 degrees? Maybe it should be a thousand degrees, maybe we should have all started off mathematics, once around is a thousand degrees. So then, half way around would be 500 degrees, and then a right angle would be 250 degrees. Why don't we do it that way? Well, you know what the answer is? It's really pretty funny. Why once around is 360 degrees?
Well because a long time ago, when people were thinking about these trigonometric functions, they were thinking one complete cycle, once around, is like going through a complete calendar year. And they thought that there were about 360 days in the calendar year, which is actually pretty close. And so therefore, they called one cycle around 360 degrees. Isn't that an utterly ludicrous, ridiculous reason to define a mathematical object by how many days there are in the year? That's the craziest, like you know going and getting a psychic reading and figuring out therefore we should call Calculus something else. I mean it's just crazy, but that's what it was, and that's how it stayed, and in fact, we're all very comfortable with that notation, including myself. But for Calculus, Calculus is a science of mathematics, where really we need careful attention to detail. It turns out that if we use that crazy artificial, 360 degrees, things get very complicated. We can make things easier if we actually use what are called radian. So I want to now give you a little reminder, crash course about radian measure of angles.
Question: What is the name of the alternative unit of angular measure mentioned in the text? Answer: Radians
Question: What is the traditional unit used to measure angles in trigonometry? Answer: Degrees
Question: According to the text, what is the name of the subject that deals with careful attention to detail in mathematics? Answer: Calculus | 677.169 | 1 |
So it all starts with a much more natural object, let's not look at a calendar, because who cares. Let's instead look at the circle. And if you look at the circle, it's beautiful, the circle is really pretty, really pretty. Now, let's suppose that we have a circle, that's radius one. So the radius of this circle is one. Sort of a standard circle, the radius is one unit. Then what is the length around the whole thing, the perimeter? Or in this case, with circles, we call it the circumference. So what's the circumference of the circle? Well, if you remember the formula, you know it's two pi r. Now here r, the radius is one, so the circumference once around is two pi, where pi is 3.14 so on. So, actually that seems to be a nice number to call one complete thing around. So let's now call, once around, two pi, and we'll call those units, radians. So radians, so once around, once around equals two pi radian. And where am I getting that, from? I'm taking a circle of radius one, I'm fixing that, standard circle, going once around and seeing how much material is required for me to go once around two pi. So I'm calling those radians.
Well, now armed with that basic idea, you can now find the radian measures of a lot of angles. For example, what's half way around? Well, that would be 180 degrees before, so what this by the way, says is in degrees, this is 360 degrees and that would equal two pi radian. Well, it would be half way around. Well, half way around would be 180 degrees, and that would be half of two pi, would be just pi. So, 180 degrees equals pi radian. And just a conversion from one unit to another, it's like metric to feet and yards, and so forth. What do we call that British or English, English? And then what about 90 degrees - well, 90 degrees would be what? Well, that would be half of the 180 so that would be half of pi, so it's pi over two. So pi over two is the radian way of saying 90 degrees, and 45 degrees would be half of that, so pi over four, and you get the idea. Now in fact, there's a way of actually converting from radian to degrees and it follows right from this fact right here. You see, because if 180 degrees equals pi radian, then what's one degree equal? Well, if I divide both sides by 180, on this side, I would see one degree and on this side, I would see pi over 180 radian. So this is a way of converting from degrees to radians. If you have five degrees, you just multiply the five by pi over 180 and you get the radian.
Question: What is the unit of measurement used for the circumference in the given context? Answer: Radians | 677.169 | 1 |
Similar, if you want to convert the other way, what you would do is divide both sides of this by pi, and see that 180 over pi degrees equals one radian. So if you want to know how many degrees is pi over six radian, you would take pi over six and multiply it by 180 over pi and you'd get how many degrees it is. These conversion uh, these conversion facts, these conversion identities, I don't think you should memorize. I think instead you should just remember once around is two pi, and that will then give you all these things, in particular this. And then you can use that to convert, by just dividing by 180 or dividing by pi, whatever is necessary.
So anyway, this is going to be the measure of angles for Calculus, and we're not going to use the degrees. And, and you may say, "Well what if I really like the degrees, can I use them anyway?" The answer is no! Calculus actually won't work that well, if you use the degrees. You really have to get accustomed to using radians. It may take a while, but once you get them, no problem. Anyway, all out of a circle of radius one.
Okay, so now armed with that basic idea, let's see if we can figure out what the graph of this looks like. Okay, what does the graph look like? We want to graph F of X equals sine x, you may remember how this goes. It's a very nice, periodic function that keeps going up and down, in a very almost hypnotic way. And here is what the graph looks like, really pretty, really pretty. So it starts off here at zero, and then it starts to go up very gradually and it gets to the highest point at one, that value there is one, that's one. And then it starts to fall; now that highest point, by the way, is achieved at pi over two, which we used to think of as 90 degrees. This is actually pi over two and then the function starts to fall very gracefully and then at zero, it crosses the x axis at pi, and it keeps coming down here, and this is three pi over two. And then it begins to rise again, and comes back to where we started, makes a complete cycle in two pi, which 360 degrees. You go once around, you make a complete cycle. So that is the sine function, that's the graph of the sine function.
Question: What is the period of the sine function in radians? Answer: 2π | 677.169 | 1 |
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