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Lesson κβ': Drawing a Tangent to a Circle
This is one of the most common constructions in technical drawing
Book III, Proposition 17 is a problem of frequent application in technical drawing: From a given point, to draw a line tangent to a given circle. At this point, we know how to draw a line through two points (Postulate 1), a line through a point parallel to a given line (I-31), or perpendicular to a given line (I-12), or a perpendicular from a given point on a line (I-11). It seems that we must find the point of tangency in order to draw a line joining it and the given point. This is what is done in Euclid's solution of the problem.
It is instructive to look at how this problem was solved when drawing by hand. Given the point and the circle, it is easy to draw the tangent by eye. This is not enough, however. The tangent point must also be found so that the point where the straight line ends and the circle begins is clear. This is necessary if the drawing is to be inked, for example. The tangent point is found by using the right angle of a triangle, as shown in the Figure. One edge of the triangle is brought parallel to the tangent that has just been drawn. A second triangle (or the T-square) is arranged so that the hypotenuse can slide along it, maintaining the direction. The triangle is then moved until the center of the circle, point E, is on the other leg. Now the point B where this side crosses the tangent can be marked.
Euclid's solution does not depend on a judgement of when a line is tangent to the circle, but finds the point of tangency B directly. Something of the sort is necessary for computer drawing as well, and the classical solutions can often be adapted, with the use of analytic geometry. In Euclid's solution, we first find the center E of the circle if necessary (III-1), then join AE. At the point Δ where this line intersects the circle, a perpendicular is drawn (I-11). A circle with radius EA is now drawn (Post. III). The perpendicular intersects the circle in Z. The line ZA then cuts the given circle at the point of tangency, B. To show that this works, we only have to prove that the angle EBA is a right angle. This is done by showing that the triangle ABE is congruent to triangle BΔE (SAS), and noting that angle EΔB is a right angle by construction.
Question: What is the method used to find the point of tangency when drawing by hand? Answer: Using a triangle, with one edge parallel to the tangent and the other leg passing through the center of the circle. | 677.169 | 1 |
To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:
Special values in trigonometric functions
There are some commonly used special values in trigonometric functions, as shown in the following table.
Inverse functions. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:
Function
Definition
Value Field
The notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "arcsecond".
Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,
These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:
Analogous formulas for the other functions can be found at Inverse trigonometric functions. Using the complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
, one can generalize all these functions to complex arguments:
Properties and applications
The trigonometric functions, as the name suggests, are of crucial importance in trigonometry
TrigonometryIn geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....
.
It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
:
or equivalently,
Question: Which trigonometric function can be defined as the integral of 1/√(1-x^2) from x to 0? Answer: Arcsine
Question: What is the Pythagorean theorem in its equivalent form? Answer: a^2 + b^2 = c^2
Question: What is the length of the shortest side in the right triangle formed by dividing an equilateral triangle of side length 1? Answer: 1/2
Question: Which theorem is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known? Answer: The law of sines | 677.169 | 1 |
20.2. We have a set of plane points Pj; these are
subject to a plane affine transformation. Show that:
(Equation 20.2)
is an affine invariant (as long as no two of i, j, k and l are the
same, and no three of these points are collinear).
Solution:
Pi is of the form (in homogeneous coordinates).
Let C denote the plane affine transformation, then
If e is an affine invariant then f should be equal to e. Where f is:
Therefore e is an affine invariant. The assumptions
that no two of i, j, k and l are the same and no three points are
collinear, are required so that det[Pi Pj
Pk] and det[Pi Pj Pl] are nonzero.
20.4. In chamfer matching, at any step, a pixel can be updated if
the distance from some or all of its neighbours to an edge are known;
Borgefors counts the distance from a pixel to a vertical or horizontal
neighbour as 3 and to a diagonal neighbour as 4 to ensure the pixel
values are integers. Why does this mean sqrt(2) is approximated as
4/3? Would a better approximation be a good idea?
Solution:
In the figure above P2 is a neighboring horizontal
pixel, and P3 is a neighboring diagonal pixel.
In terms of exact math and using Pythagoras' Theorem:
Borgefors approximates P1P2 as 3 and
P1P3 as 4, for integer arithmetic. So
is approximated as 4/3, (i.e,
sqrt(2) is approximated as 4/3).
A better approximation is P1P2 as 10 or 5
(instead of 3), and P1P3 as 14 or 7 (instead of 4).
24.1. Assume that we are delaing with measurements x in some
feature space S. There is an open set D where any element is
classified as class one, and any element in the interior of S-D is
classified as class two.
(a) Show that
(b) Why are we ignoring the boundary of D (which is the same as the
boundary of S-D) in computing the total risk?
Solutions:
(a) The probability that given x it is of class 1 but lies in S-D
is = P{1->2 | using s}
The probability that given x it is of class 2 but lies in D is = P{2->1 | using s}
Therefore,
(b)The boundary is a break-even point. The type I and type II
errors both give the same risk, so we don't have any preference over
comitting type I or type II error. This means that if a point lies on
the boundary, we can classify it as belonging to either class 1 or
class 2. We have no preference, so we ignore the boundary of D in
computing the total risk.
Question: What is the form of Pi in homogeneous coordinates? Answer: Pi is of the form (in homogeneous coordinates). | 677.169 | 1 |
Alen94 wrote:They told me that the contraction angle shoud be around 15 degrees to ensure flow similarity. Now I asked them how to reason this number. Does anybody of you have an idea?
The design you pictured is symetric, so if each angle measures 15 degrees, you actually have 30 degress of narrowing. Trigonmetry Tangent of an angle is the ratio of the opposite (diameter) over the adajacent (length of transition) side of a right triangle. If you want the transition to be at least 3 times the final diameter, tan x = 1/3 results in x = 18 degrees.
But lets look at your fan source. Are you planning on using a simple 30" square box window fan? If yes, then the motor area in the middle is not moving air and the four corners are not moving air. You are just going to have to build one and measure the pressues and speeds and see how even they turn out. If the flow isn't laminar enough, you can always lengthen the straight section.
Question: What is the ratio of the transition length to the final diameter for the design to work properly? Answer: At least 3:1 | 677.169 | 1 |
The diagram shows that the cut through the side of the bagel rotates a full 360 degrees as you go around the bagel. This gives two interlocking rings when you try to take it apart, but neither is a Mobius band since the flat surface of either rotates 360 degrees as you go around it (ie, it has two half-twists, not one as in a standard Mobius band). But if you rotate the original cut only 180 degrees as you go around the bagel, you get a double length ring that doesn't separate into two parts (it's like two coils of a spring with the ends welded together), and it has two half-twists.
Take a Mobius band and cut it lengthwise (down the middle of the strip). Because of the half-twist in the original band, the two halves cross over and connect to each other, so you get a double-length band with three half-twists.
Try cutting the strip into thirds lengthwise. The outer two parts are like the two halves of the above example, so they from a double length band with three half-twists, while the middle part is just a thinner version of the original band. And these two parts are interlinked, but not as simply as in the bagel example.
Cut a Klein bottle into two symmetric halves, and you get two separate Mobius bands
Question: What is the rotation angle for the second described cut through the bagel? Answer: 180 degrees | 677.169 | 1 |
Why Learn / Use Identities?
Identities (in any branch of mathematics) help us to:
solve
simplify
or gain insight into
mathematical problems.
Identities are a lot like synonyms inWhile I call this advanced, it does not mean harder or more complicated, it just means more abstract. Understanding the trigonometric functions (sine, cosine, tangent) using right angle triangles is simply a special case of trigonometric functions using the unit circle.
I strongly recommend first reading and understanding the article Understanding Sine, Cosine, and Tangent first, because it explains the history and reasoning behind the trigonometric functions.
This is the way trigonometric functions are generally understood and defined in mathematics.
Trigonometric functions were originally developed and understood from the study of right angle triangles. The problem with using right angle triangles is that trigonometric functions can only be defined for angles between 0° and 90°, but not for angles ≤ 0° or ≥90° because no such right angle triangles exist.
The Unit Circle
The Unit … Read entire article »
An article explaining trigonometric functions using the unit circle can be found here
Using the unit circle is the standard way trigonometric functions are defined and understood in mathematics.
I recommend reading and understanding this article first. Later, if you want to understand how trigonometric functions are defined for values greater than 90° or less than 0°, go and read the other article.
Sine is often introduced as follows:
Which is accurate, but causes most people's eyes to glaze over.
The problem is that from the time humans starting studying triangles until the time humans developed the concept of trigonometric functions (sine, cosine, tangent, secant, cosecant and cotangent) was over 3000 years.
A Little History
The ancients studied triangles. One of the things they did was to compare the lengths of the sides of triangles:
A triangle has … Read entire article »
This tutorial examines the concept of the average as a single value representing a collection of values. It focusses on the mean, median, and mode.
The average (especially in physics) can also mean the center or balance point, but, for most everyday use, we tend to think of the average as representative value.
Average comes from the Old French avarie which came from the Old Italian avaria which came from the Arabic awariyah meaning damaged goods or merchandise. Which is probably apt, given how how badly averages are often misused.
The Old French avarie used to mean the damage sustained to a ship or its cargo. The meaning later shifted to mean an equal distribution of the costs of such damage. For example, if ten men pooled money together and hired a ship … Read entire article »
This is illustrated in the image below:This is true for the other two unshaded interior angles. It is also true for the alternate exterior angles (but not proved here).
Axioms
Question: What is the standard way trigonometric functions are defined and understood in mathematics? Answer: Using the unit circle.
Question: Are trigonometric functions originally defined using right angle triangles? Answer: No, the original definition of trigonometric functions is not using right angle triangles.
Question: Which of the following is NOT a type of average? A) Mean B) Median C) Mode D) Mode of the Mean Answer: D) Mode of the Mean
Question: What are some benefits of using identities in mathematics? Answer: Identities help to solve, simplify, or gain insight into mathematical problems. | 677.169 | 1 |
It sounded from your description like you were trying to define Spherical co-ordinates, a 3D system of TWO angles and a distance. The two angles correspond to lattitude and longitude on a globe. Spherical co-ordinates is one of two common 3D polar systems, the other is cylindrical co-ordinates, a system of two distances and an angle.
There is a 3D system that uses three angles. It involves two predefined reference points on a predefined reference plane. The three angles are the polar direction from each of the reference points and the angle from the plane. A 3D system that involved THREE angles and a distance would be overconstrained
Question: How many distances are there in cylindrical coordinates? Answer: Two | 677.169 | 1 |
root extraction. Finding a number that can be used as a factor a given number of times to produce the original number; for example, the fifth root of 32 = 2 because 2 x 2 x 2 x 2 x 2 = 32).
rotation. A rotation in the plane through an angle q and about a point P is a rigid motion T fixing P so that if Q is distinct from P, then the angle between the lines PQ and PT(Q) is always q . A rotation through an angle q in space is a rigid motion T fixing the points of a line l so that it is a rotation through q in the plane perpendicular to l through some point on l.
scalar matrix. A matrix whose diagonal elements are all equal while the nondiagonal elements are all 0. The identity matrix is an example.
scatterplot. A graph of the points representing a collection of data.
scientific notation. A shorthand way of writing very large or very small numbers. A number expressed in scientific notation is expressed as a decimal number between 1 and 10 multiplied by a power of 10 (e.g., 7000 = 7 x 103 or 0.0000019 = 1.9 x 10-6 ).
similarity. In geometry, two shapes R and S are similar if there is a dilation D (see the definition of dilation) that takes S to a shape congruent to R. It follows that R and S are similar if they are congruent after one of them is expanded or shrunk.
sine. Sin(q) is the y- coordinate of the point on the unit circle so that the ray connecting the point with the origin makes an angle of q with the positive x- axis. When q is an angle of a right triangle, then sin(q) is the ratio of the opposite side with the hypotenuse.
square root. The square roots of n are all the numbers m so that m2 = n. The square roots of 16 are 4 and -4. The square roots of -16 are 4 i and -4 i .
standard deviation. A statistic that measures the dispersion of a sample.
symmetry. A symmetry of a shape S in the plane or space is a rigid motion T that takes S onto itself (T(S) = S). For example, reflection through a diagonal and a rotation through a right angle about the center are both symmetries of the square.
system of linear equations. Set of equations of the first degree (e.g., x + y = 7 and x - y = 1 ). A solution of a set of linear equations is a set of numbers a, b, c, . . . so that when the variables are replaced by the numbers all the equations are satisfied. For example, in the equations above, x = 4 and y = 3 is a solution.
translation. A rigid motion of the plane or space of the form X goes to X + V for a fixed vector V.
Question: In a rotation in the plane, what remains fixed? Answer: The point P
Question: What is the purpose of a scatterplot? Answer: To graphically represent a collection of data points. | 677.169 | 1 |
1. Take a
line, horizontal and straight. Split it into 3 equal 3 inch
parts, and take out the middle segment. Replace it with an
equilateral triangle, and make sure each side is 3 in. long.
Your curve should now look like what is shown below. Note
that with the pictures on this page, the segments are not 3
in. long. They are smaller-size pictures.
2. Each time
you go through that loop, or calculation, is 1
iteration.
Now, repeat the process again, but do it to all the new
lines you have. If you are stuck, refer to the picture
below.
3. Do the
same process as many more times as you wish. If you repeat
the process 4 times, it may look like the image
below.
Question: What is the length of each segment when the line is split into three equal parts? Answer: 3 inches | 677.169 | 1 |
An Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts by translations, or, conversely, an Euclidean vector is the difference (displacement) in an ordered pair of points, not a single point. Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere. When certain point is chosen, it can be declared the origin and subsequent calculations may ignore the difference between a point and its coordinate vector, as said above. See point–vector distinction for details.
Euclidean structure[
These are distances between points and the angles between lines or vectors, which satisfy certain conditions (see below), which makes a set of points an Euclidean space. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on Rn. The inner product of any two real n-vectors x and y is defined by
where xi and yi are ith coordinates of vectors x and y respectively. The result is always a real number.
This distance function (which makes a metric space) is sufficient to define all Euclidean geometry, including the dot product. Thus, a real coordinate space together with this Euclidean structure is called Euclidean space. Its vectors form an inner product space (in fact a Hilbert space), and a normed vector space.
Angle[
The (non-reflex) angleθ (0° ≤ θ ≤ 180°) between vectors x and y is then given by
where arccos is the arccosine function. It it useful only for n > 1,[1] and the case n = 2 is somewhat special. Namely, on an oriented Euclidean plane one can define an angle between two vectors as a number defined modulo1 turn (usually denoted as either 2π or 360°), such that ∠yx = −∠xy. This oriented angle equals either to the angle θ from the formula above or to −θ. If one non-zero vector is fixed (such as the first basis vector), then each non-zero vector is uniquely defined by its magnitude and angle.
Unlike the aforementioned situation with distance, the scale of angles is the same in pure mathematics, physics, and computing. It does not depend on the scale of distances: all distances may be multiplied to some fixed positive factor, while all angles preserve. Usually the angle is considered as a dimensionless quantity, but there are different units of measurement, such as radian (preferred in pure mathematics and theoretical physics) and degree ° (preferred in most applications).
Rotations and reflections[
Symmetries of an Euclidean space are transformations which preserves the Euclidean metric (called isometries). Although aforementioned translations are most obvious of them, they have the same structure for any affine space and do not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed, which may be seen as the origin without loss of generality. All transformations, which preserves the origin and the Euclidean metric, are linear. Such transformations Q must, for any x and y, satisfy:
Question: What is the origin in Euclidean space? Answer: The origin is a chosen point in the space, which can be translated anywhere.
Question: What are the transformations that preserve the origin and the Euclidean metric called? Answer: They are called linear transformations.
Question: What is the effect of multiplying all distances in Euclidean space by a fixed positive factor? Answer: All angles preserve, but the scale of distances changes. | 677.169 | 1 |
Groups SO(n) are well-studied for n ≤ 4. There are no non-trivial rotations in 0- and 1-spaces. Rotations of an Euclidean plane (n = 2) are parametrized by the angle (modulo 1 turn). Rotations of a 3-space are parametrized with angle and axis, whereas a rotation of a 4-space is a superposition of two 2-dimensional rotations around perpendicular planes.
This structure of Euclidean group essentially explains to which conditions should a metric space satisfy to be an Euclidean space. First, it must be translationally-invariant with respect to some (finite-dimensional) real vector space, which means that the space itself is an affine space. One may understand it as that the space is flat, not curved, and points do not have different properties because any point can be translated to any another point. Second, the metric must correspond in the aforementioned way to some positive-defined quadratic form on this vector space, because point stabilizers have to be isomorphic to O(n).
Non-Cartesian coordinates[
Cartesian coordinates are arguably the standard, but not the only possible option for an Euclidean space. Skew coordinates are compatible with the affine structure of En, but make formulae for angles and distances more complicated.
Any two distinct point lie on exactly one line. Any line and a point outside it lie on exactly one plane. These properties are studied by affine geometry, which is more general that Euclidean one, and can be generalized to higher dimensions.
Line segments and triangles[
The sum of angles of a triangle is an important problem, which exerted a great influence to 19th-century mathematics. In an Euclidean space it invariably equals to 180°, or a half-turn
This is not only a line which a pair (A, B) of distinct points defines. Point of the line which lie between A and B, together with A and B themselves, constitute a line segmentAB. Any line segment has the length, which equals to distance between A and B. If A = B, then the segment is degenerate and its length equals to 0, otherwise the length is positive.
A (non-degenerate) triangle is defined by three points not lying on the same line. Any triangle lies on one plane. The concept of triangle is not specific to Euclidean spaces, but Euclidean triangles have numerous special properties and define many derived objects.
A triangle can be thought of as a 3-gon on a plane, a special (and the first meaningful in Euclidean geometry) case of a polygon.
Polytopes and root systems[
Platonic solids are the five polyhedra which are most regular in combinatoric sense, but also, their symmetry groups are embedded into O(3).
Polytope is a concept which generalizes polygons on a plane and polyhedra in 3-dimensional space (which are among the earliest studied geometrical objects). A simplex is a generalization of a line segment (1-simplex) and a triangle (2-simplex). A tetrahedron is a 3-simplex.
Alternatives and generalizations[
Question: What is the sum of angles of a triangle in an Euclidean space? Answer: The sum of angles of a triangle in an Euclidean space invariably equals to 180°, or a half-turn.
Question: What is one of the conditions a metric space must satisfy to be considered an Euclidean space? Answer: The metric must correspond to some positive-defined quadratic form on a finite-dimensional real vector space.
Question: What is a generalization of a line segment in higher dimensions? Answer: A simplex is a generalization of a line segment (1-simplex) in higher dimensions.
Question: What is the parameterization for rotations in a 2-dimensional Euclidean space? Answer: Rotations in a 2-dimensional Euclidean space are parametrized by the angle (modulo 1 turn). | 677.169 | 1 |
Crossings vs. angle, 1,000,000 needles
By the time we get to a million points, estimates are usually accurate to
three
decimals. The program will run 100 million needles in a few seconds,
but accuracy increases quite slowly.
Now, if you think you understand all of this, try to explain it some
someone else!
Question: What is the total number of points used in the comparison? Answer: 1,000,000 | 677.169 | 1 |
This method returns the coordinates of the intersection of 2 lines. Test the line you want to draw against EACH side of the square and if the length of the array returned is 2 then use the intersection co-ordinates as the line end.
/**
* Find the point of intersection between two lines. <br>
* An array is returned that contains the intersection points in x, y order.
When working with code, it can easily become very difficult to read... Try using Edit > Auto Format, in processing to make your code easier to read. No offense, but your code is a little hard to read....
Question: What does the returned array represent? Answer: The returned array contains the intersection points in x, y order. | 677.169 | 1 |
'National Flags' printed from
National Flags
What shapes can you see in it? Can you describe them and their angles?
Does the flag have any lines of reflective symmetry, if so how many lines?
Can you find any pairs of parallel lines? If so mark them on your flag.
Are there any lines perpendicular to one another?
Can you find a way to classify the shapes in your flag?
Now try with another flag.
This problem was developed for us by Claire Willis.
Why do this problem?
This problem gives opportunities for children to identify, visualise and describe the characteristics and properties of 2 D shapes in the context of a meaningful real life setting. It also provides experience of classifying and measuring angles and identifying lines of symmetry.
Possible approach
With the children working in pairs ask one person to pick a 2 D shape from a set of shapes and describe it to their partner without them seeing it. The second person must then draw the shape they think it is.
Next let the children choose their own flag and find ways in which to investigate it. Templates of the flags can be downloaded here to enable the children to mark and measure angles, and identify parallel and perpendicular lines. Mirrors and tracing paper would be useful. Here is a useful website that gives lots of background information about flags and printable resources.
Key questions
What shapes can you see in your flag? Are they regular or irregular?
Can you describe their angles? Can you estimate them? Measure them?
Does the flag have any lines of reflective symmetry?
Can you find any parallel and/or perpendicular lines?
Possible extension
Children could look at the order of rotational symmetry of their flags, and work out areas of different shapes given certain dimensions.
Question: Can you describe the angles of the triangle in the flag? Answer: The triangle in the flag has one right angle (90 degrees) and two acute angles, which are less than 90 degrees. | 677.169 | 1 |
Critical Reasoning
Math (DS)
If vertices of a triangle have coordinates , what is the area of the triangle?
1.
2. angle at the vertex equals 90 degrees
Question Discussion & Explanation
Correct Answer - A - (click and drag your mouse to see the answer)
GMAT Daily Deals
Veritas Prep:...
Math (PS)
Which of the following sets has the same standard deviation as set ?
(A)
(B)
(C)
(D)
(E)
Question Discussion & Explanation
Correct Answer - E - (click and drag your mouse to see the answer)
GMAT Daily Deals
e-GMAT: Save up to $595 on the world's...
Math (PS)
Two consultants can type up a report in 12.5 hours and edit it in 7.5 hours. If Mary needs 30 hours to type the report and Jim needs 12 hours to edit it alone, how many hours will it take if Jim types the...
Math (PS)
There is a certain triangle with sides 7, 10, and . If it is known that is an integer, how many different values are there of ?
(A) 8
(B) 10
(C) 13
(D) 14
(E) 16
Question Discussion & Explanation
Correct Answer - C - (click and...
Math (PS)
A steamer going upstream would cover the distance between town A and town B in 4 hours and 30 minutes. If the same steamer going downstream would cover the distance between the towns in 3 hours, how long would it take a raft moving...
Math (DS)
There are two schools in the village. The average age of pupils in the first school is 12.2 years; the average age of pupils in the second school is 13.1 years. What is the average age of all school pupils in the village?
1. There...
Question: Is the angle at the vertex (0,0) in the given triangle a right angle? Answer: Yes
Question: If there are 100 pupils in the first school and 150 in the second, what is the total number of pupils in the village? Answer: 250 pupils | 677.169 | 1 |
If triangle above is congruent to triangle (not shown), which of the following must be the length of one side of triangle ?
Answer Choices
(A)
(B)
(C)
(D)
(E) It cannot be determined from the information given.
Yep! That's right.
Explanation
Triangle is congruent to triangle , so the lengths of the three sides of triangle are the same as the lengths of the three sides of triangle . Triangle is a - - triangle with hypotenuse of length , so the other two sides of triangle have lengths and . Therefore, the lengths of the sides of triangle must be , , and . Of the choices given, only is one of these values
Question: Which of the given answer choices is the length of one side of triangle? Answer: (C) | 677.169 | 1 |
And Why
New Vocabulary
Building Proofs in the Coordinate Plane
In Lesson 5-1, you learned about midsegments of triangles. A trapezoid also has a midsegment. The midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides. It has two unique properties.
Key Concepts
Theorem 6-18 Trapezoid Midsegment Theorem
(1) The midsegment of a trapezoid is parallel to the bases. (2) The length of the midsegment of a trapezoid is half the sum of the lengths of the bases.
|| , || , and MN = (TP + RA).
Formulas for slope, midpoint, and distance are used in a proof of Theorem 6-18.
Planning a Coordinate Geometry Proof
Developing Proof Plan a coordinate proof of Theorem 6-18.
Given is the midsegment of trapezoid TRAP.
Prove || , || , and MN = (TP + RA).
Plan Place the trapezoid in the coordinate plane with a vertex at the origin and a base along the x-axis. Since midpoints will be involved, use multiples of 2 to name coordinates. To show lines are parallel, check for equal slopes. To compare lengths, use the Distance Formula.
Real-World Connection
The rectangular flag at the left is constructed by connecting the midpoints of its sides. Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of the sides of a rectangle is a rhombus.
Given MNPO is a rectangle. T, W, V, U are midpoints of its sides.
Prove TWVU is a rhombus.
Plan Place the rectangle in the coordinate plane with two sides along the axes. Use multiples of 2 to name coordinates. A rhombus is a parallelogram with four congruent sides. From Lesson 6-6, Example 2, you know that TWVU is a parallelogram. To show , use the Distance Formula.
Coordinate Proof: By the Midpoint Formula, the coordinates of the midpoints are T(0, b), W(a, 2b), V(2a, b), and U(a, 0). By the Distance Formula,
Question: What are the two unique properties of the midsegment of a trapezoid? Answer: (1) It is parallel to the bases. (2) Its length is half the sum of the lengths of the bases.
Question: What formulas are used in the proof of this theorem? Answer: Formulas for slope, midpoint, and distance. | 677.169 | 1 |
GEO Thank you for using the Jiskha Homework Help Forum. 1. Since Lisboa (Lisbon) is the capital of Portugal, you need Yaoundé which is the capital of Cameroon. 2. Caribbean. If this is a class of GEO, or geography, of course this will make no sense without an atlas, a globe...
Wednesday, October 22, 2008 at 6:11pm by SraJMcGin
Statistics - can someone check me P(geo ...
Saturday, February 12, 2011 at 5:33pm by Rachal
geometry If you double one dimension, halve another, and keep the third the same, wouldn't the volume be identical?
Friday, June 25, 2010 at 12:33pm by Writeacher
Geology I Googled earthquake waves and found several sites -- all with the same answer. This seems to be the best with the clearest explanation.
Thursday, December 6, 2007 at 5:55pm by Ms. Sue
Math I get: ...
Saturday, February 12, 2011 at 9:05pm by Rachal
math You seem to be missing one measurement. For volume, you need the width, length, and height.
Thursday, January 22, 2009 at 10:30pm by Ms. Sue
Math This site has a great explanation and diagrams of congruent triangles.
Wednesday, September 5, 2007 at 7:35pm by Ms. Sue
Nautical Studies Here in North America: Boston, Los Angeles Harbor, Adak Alaska. Other charts are available, see the links at the top.
Wednesday, October 31, 2007 at 6:17am by bobpursley
geometry
Monday, February 16, 2009 at 2:35pm by Damongeometry how do you find out the center of a circle that circumscribes a triangle? See the midperpendicular paragraph.
Tuesday, November 14, 2006 at 3:45pm by jessica
math I need 2 know how 2 find the measure of a regular haxagon!! HELP ME PLEASE!! This site explains how to find the area of a regular hexagon.
Monday, February 19, 2007 at 4:50pm by Sweetpea:)
Math Yep. It's either A or C. Is angle C a vertex or base angle?
Tuesday, February 26, 2013 at 5:38pm by Ms. Sue
World geography
Friday, April 26, 2013 at 9:01pm by Ms. Sue
Question: What is the name of the person who posted about finding the measure of a regular hexagon? Answer: Sweetpea:)
Question: What is the name of the person who posted about congruent triangles? Answer: Ms. Sue | 677.169 | 1 |
hyperbola
Geom. the path of a point that moves so that the difference of its distances from two fixed points, the foci, is constant; curve formed by the section of a cone cut by a plane more steeply inclined than the side of the cone
A plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone. It is the locus of points for which the difference of the distances from two given points is a constant.
Origin:
Origin: New Latin
Origin: , from Greek huperbolē, a throwing beyond, excess (from the relationship between the line joining the vertices of a conic and the line through its focus and parallel to its directrix); see hyperbole
Question: What is the relationship between the line joining the vertices of a conic and the line through its focus and parallel to its directrix in the context of a hyperbola? Answer: The relationship between them is what gives the hyperbola its name, as 'hyperbola' comes from the Greek word for excess or throwing beyond. | 677.169 | 1 |
In this two-day lesson, students will collaborate to create a healthy pizza using only geometric items that have been precisely measured. Students must identify the items as triangle, quadrilateral (parallelogram), or cube. Next, students will measure the items that they place on their pizza. Finally, students will bake their pizza; therefore, having their math and eating it, too!Mathematics and Statistics2013-04-04T13:40:18Course Related MaterialsQuilting With My Pal, Pythagoras!
In this unit lesson, students will learn about the Pythagorean Theorem and how it is evident in our everyday world. Students will apply the concept of the Pythagorean Theorem to the squares of a quilt. Each quilt square will be designed and crafted by a student. In the end, a beautiful quilt will be made with the help of Pythagoras himself!Mathematics and Statistics2013-04-04T13:40:17Course Related MaterialsA Triangle to Remember
This lesson will introduce students to different types of triangles by using a clip from the movie A Walk to Remember. They will take part in interactive tutorials to strengthen the concepts of classifying triangles. The students will then use features on Microsoft Word to create an interactive triangle poster and show what they know.Mathematics and Statistics2013-04-04T13:40:16Course Related MaterialsGeometric Man
The students will explore angles, rays, line segments, perpendicular lines, parallel lines, and two-dimensional figures through the use of the book "Shape Up! Fun With Triangles and Other Polygons" by David A. Adler. They will also classify quadrilaterals through an interactive website.Mathematics and Statistics2013-04-04T13:40:14Course Related MaterialsConstruct This!
Students will construct equilateral triangles, squares, and regular hexagons inscribed in circles using the free GeoGebra computer program.Mathematics and Statistics2013-04-04T13:40:09Course Related MaterialsTriangles and Things
The Greedy Triangle, a Marilyn Burns book, will assist students in determining the simple relationships among polygons. With a wish, the Greedy Triangle is transformed into various polygons. With the addition of a side and an angle, students will be able to determine the difference among triangles, quadrilaterals, pentagons, and hexagon. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.ArtsHumanitiesMathematics and Statistics2013-04-04T13:40:06Course Related MaterialsI Can Determine The Height Of A Rocket!
Question: Which mathematical concept is the focus of the second lesson? Answer: The Pythagorean Theorem | 677.169 | 1 |
The lesson is intended to give students a fun real-world experience in applying their math skills. They will use trigonometric ratios to calculate heights of tall structures. They will also use the Internet to convert their calculations from standard to metric units and visa versa.Mathematics and StatisticsScience and Technology2013-03-26T22:52:03Course Related MaterialsRight Triangles Inscribed in Circles II
The result here complements the fact, presented in the task ``Right triangles inscribed in circles I,'' that any triangle inscribed in a circle with one side being a diameter of the circle is a right triangle. A second common proof of this result rotates the triangle by 180 degrees about M and then shows that the quadrilateral, obtained by taking the union of these two triangles, is a rectangle.Mathematics and Statistics2013-03-15T14:03:50Course Related MaterialsWhy Does SSS Work?
This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection. There are many other ways in which to exhibit the congruency and students and teachers are encouraged to explore the different possibilities.Mathematics and Statistics2013-03-15T14:03:50Course Related MaterialsWhy Does SAS Work?
For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.Mathematics and Statistics2013-03-15T14:03:49Course Related MaterialsWhy Does ASA Work?
The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.Mathematics and Statistics2013-03-15T14:03:49Course Related MaterialsWhen Does SSA Work to Determine Triangle Congruence?
The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.Mathematics and Statistics2013-03-15T14:03:49Course Related MaterialsReflections and Isosceles Triangles
Question: What is the least number of reflections required to exhibit the congruence between two triangles that share a side, as mentioned in the "Why Does ASA Work?" material? Answer: Only one rigid transformation is required to exhibit the congruence between two triangles that share a side. | 677.169 | 1 |
Because lines l and m are
parallel and line AB is a transversal, the angle
whose measure is labeled as 120º is supplementary to .
Now we have a 30-60-90 triangle whose longer leg, AC,
is also the distance between lines l and m.
Using the :: side ratios for 30-60-90 triangles
you can use the hypotenuse length to calculate the lengths of the
other two legs. The short leg has a : ratio to the hypotenuse, so its
length is 5/2.
The long leg has a : ration to the short leg, so its
length is 5⁄2.
2. E
Let's analyze each statement separately.
This statement implies that DF = EF.
We know that BF = AF because it
is given that line CF is the perpendicular bisector
of AB, and by definition, F is the
midpoint of AB. It is also given that the area
of triangle CDB is equal to the area of triangle CEA.
These two triangles share the same height, and since the area of
a triangle is found by the formula 1⁄2bh, it follows that
if their areas are equal, their bases are equal too. If BD
= AE, then by subtracting DE from each
segment, we have BE = AD and thus EF =
DF. So statement I is true.
This
statement is simply not backed by any evidence. All we know is that BE = AD, EF = DF,
and BF = AF. As long as points E and D are
equidistant from F, all these conditions hold, so there is no guarantee
that they are the midpoints of BF and AF,
respectively. E and D could be
anywhere along BF and AF, respectively,
as long as they are equidistant from F. Thus, this statement
is not necessarily true.
Triangles CDB and CEA are
equal in area; this is given. By subtracting the area of triangle CED from
each of these triangles, we see that triangles CEB and CDA must
have the same area. This statement is true.
Only statements I and III must be true.
3. C
The area of a triangle with base x and
height h is given by the formula 1⁄2xh.
The area of a square with sides of length x is x2.
Since you know the two shapes have equal areas, you can set the
two expressions equal to each other and solve for h:
The correct answer is h = 2x.
4. D
If ABD is an equilateral triangle,
then AD = AB = BD =
4, and all the sides of the rhombus have a length of 4 (by definition
of a rhombus, all sides are congruent). Also, by definition of a
rhombus, opposite angles are congruent, so . Draw an altitude from a to DC to
Question: Are the statements I, II, and III in the text all true? Answer: No, only I and III are true.
Question: What is the length of the short leg of the 30-60-90 triangle if the hypotenuse is 10? Answer: 5 | 677.169 | 1 |
As you can see from the excel file the measured interior angles of Korman added up to exactly 1080 degrees. This means that our angles do form the octagonal shape of the building. Our next step was to check the side lengths. We calculated the latitude and departure of each side and found that the GPS measurements were very flawed. They showed that the latitude was off by 100 feet and the departure was off by 50 feet! Considering the relatively small size of the building this flaw is enormous. However, we still went through the compass correction formula and adjusted the side lengths to close the traverse. The final values of angles and length are shown in the adjusted measurements file.
Conclusion
As stated previously we found that the latitude and departure of our measurements were quite erred. We think that this may be due to the fact that we were measuring small distances. When the GPS device is used in cars or travel it is usually over a distance of miles, this means that an error of 50 ft will have less effect on results. However, since our project required the measurement of 50 ft sides this kind of error could potentially double the correct value. The angles on the other hand added up to 1080 as expected. Although we think this may be coincidence since some geometrically corresponding angles are not the same. Also, we found that unless you read the GPS manual it is very difficult to figure out how to measure things. The "page" button is almost misleading in that instead of making a new page of data it cycles through the functions of the device. To reset data you must hit "menu" then "reset data" then select the data you wish to reset before pressing "enter" one last time. We felt that this was very monotonous and time consuming especially since we had to do this after each of the 8 sides. Altogether the 'mini' lab taught us the general operations of the device as well as demonstrated the variability of its results.
Conclusion
what we learned... what we already knew... what we can conclude from our miniprojects
Question: What shape did these angles form? Answer: An octagonal shape | 677.169 | 1 |
Midpoint and Distance Formulas
In the new section on Conic Sections, Dr. Eaton first begins with the Midpoint Formula and Distance Formula. After describing the formulas and several examples, you are able to work on your own with four additional exercises to make sure you can use both formulas appropriately.
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Midpoint and Distance Formulas
Make sure that you know and understand how to use both of these formulas. They will be used in a lot of later work.
Remember that the order of the x coordinates in the distance formula does not matter. So take whichever difference is easier to compute. The same comment applies to the y coordinates.
After squaring the differences in the distance formula, be sure to take the positive square root of their sum.
Midpoint and Distance Formulas
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Question: In which section of the course are these formulas introduced? Answer: Conic Sections | 677.169 | 1 |
You can put this solution on YOUR website!***
The earth makes one revolution every 24 hours.
Each revolution=2π*radius=7920π miles
linear speed of earth=7920π miles/24 hours≈1037 mi/hr
In order to make it appear the sun stays in the same position, one must travel at this speed against the rotation of the earth which is west to east. Traveling at this speed east to west at the equator in effect makes one stationary with respect to the sun. I think.
Trigonometry-basics/720955: 1 solutions Answer 442158 by lwsshak3(6522) on 2013-03-02 18:37:04 (Show Source):
You can put this solution on YOUR website!
***
Basic equation for sin function: y=ASin(Bx-C), A=amplitude, Period=2π/B, phase shift=C/B.
For given equation: y=1/2 sin3x
Amplitude=1/2
B=3
Period=2π/B=2π/3
C=0
phase shift: none
Because there is no phase shift, the graph is like the basic sin curve with amplitude=1/2 and period=2π/3
As with the basic sin curve, it intersects the x-axis at zero, half the period, and at the end of the period. So, for given function, points of intersection for one period are: (0,0), (π/3,0) and (2π/3,0)
You can put this solution on YOUR website! Find the exact value of x between 0 degrees and 180 degrees for which each is true.
cos x = sqrt(-3)/2
I believe this should be written:cosx=-√3/2
For given domain, [0,180º], reference angle is in quadrant II where cos<0.
cosx=-√3/2
x=150º
You can put this solution on YOUR website! How can do you prove that
secx - (tanx)(sinx) = cosx?
Start with left side:
secx-(tanx)(sinx)
=(1/cosx)-(sinx/cosx)sinx
=(1/cosx)-sin^2x/cosx
=(1-sin^2x)/cos)
=cos^2x/cosx
=cosx
verified: left side=right side
You can put this solution on YOUR website! Find the coordinates of the vertex,focus,ends of the latus rectum and the equation of the directrix.Draw the parabola of:
x^2=12(y+7)
...
This is an equation of a parabola that opens upwards:
Its basic form of equation:(x-h)^2=4p(y-k)
For given equation: x^2=12(y+7)
Question: What is the Earth's linear speed in miles per hour? Answer: Approximately 1037 mi/hr.
Question: In which direction should one travel to appear stationary with respect to the sun? Answer: East to west. | 677.169 | 1 |
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,
then the derivatives will scale by amplitude.
.
Here, k is a constant that represents a mapping between units. If x is in degrees, then
.
This means that the second derivative of a sine in degrees satisfies not the differential equation
,
but rather
;
cosine's second derivative behaves similarly.
This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.
Identities
Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is always 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity reads,
,
which is more commonly written with the exponent "two" next to the sine and cosine symbol:
.
In some cases the inner parentheses may be omitted.
Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to Ptolemy; one can also produce them algebraically using Euler's formula.
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.
These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers in a sum of numbers and greatly speed operations, much like the logarithm function.
Calculus
For integrals and derivatives of trigonometric functions, see the relevant sections of table of derivatives, table of integrals, and list of integrals of trigonometric functions. Below is the list of the derivatives and integrals of the six basic trigonometric functions.
Definitions using functional equations
In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the sum and difference formulas. Taking as given these formulas and the Pythagorean identity, for example, one can prove that only two real functions satisfy those conditions. Symbolically, we say that there exists exactly one pair of real functions and such that for all real numbers x and y, the following equations hold:
with the added condition that
.
Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers. As an example, this derivation can be used to define trigonometry in Galois fields.
Computation
Question: What are the sum and difference formulas used for? Answer: The sum and difference formulas give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves.
Question: Which trigonometric identity states that the square of the sine plus the square of the cosine is always 1? Answer: The Pythagorean identity.
Question: What are the double-angle formulas? Answer: The double-angle formulas are simpler equations derived from the sum formulas when the two angles are equal.
Question: What happens to the derivatives of sine and cosine when their arguments in radians are scaled by frequency? Answer: The derivatives scale by amplitude. | 677.169 | 1 |
Weekly Photo Challenge: Geometry
No, that isn't a spaceship landing pad, it's actually a luxury hotel and resort called Marina Bay Sands in Singapore. However, That isn't the geometry I'm submitting for this week's Photo Theme. The footbridge leading across the water to the Marina Bay Sands is called the Helix Bridge and it stands as quite an architectural statement. Because of it's curved frame, the Helix Bridge is the world's first curved bridge.
As the name suggests, it's modeled after a DNA helix structure. At night, lights around the structure illuminate in four colours, indicating the four base pair chemicals of DNA, Adenine Guanine, Cytosine, and Thymine.
On one side of the bridge, the one being faced in this picture, is the Marina Bay Sands resort with its rooftop park, bar, and restaurant and a very upscale shopping mall whose interior can be traversed by a Venetian-style riverboat.
On the side the photo is taken on is a Ferris wheel, a floating football field and an array of multi-coloured bleachers along a nice riverwalk.
Question: How many base pair chemicals are indicated by the four colors of lights on the bridge at night? Answer: Four. | 677.169 | 1 |
Theorem 3.4:
(Pythagoras)1 Let f, d, and r be the three sides of a right triangle, r the hypoteneuse.
Then
a2 + f2 = r2
Theorem 3.5:
(The Triangle Inequality) Let A, B, and C be three points in the
plane. Then the distance from A to B is less than or equal to the sum
of the distances from A to C and B to C with equality if and only if C is on the line segment between A and B.
Theorem 3.14: If two circles have the same radius, then the points of intersection
between the two circles lie on the perpendicular bisector of the line
segment joining the two centers.4
Theorem 3.15: Given a circle with center (x0, y0) and radius r and a point (x1, y1),
if the point is outside the circle, there are exactly two lines through the point tangent to the circle, if the point is on the circle, there is exactly one line through the point tangent to the circle and it is perpendicular to the line joining the point to the center of the circle, and if the point is inside the circle, there are no lines through the point tangent to the circle.
Theorem 3.16: Given a point A not on a line, any point whose distance from A is
less than the distance from A to its foot in the line is on the same side of the line as A.
Question: If a point is outside a circle, how many lines through that point are tangent to the circle? Answer: There are exactly two lines through the point tangent to the circle.
Question: Is the Pythagorean theorem (Theorem 3.4) about the relationship between the sides of a right triangle? Answer: Yes | 677.169 | 1 |
Question 23753: A triangle has vertices A(-1,k) B(6,k-1) and C(2,-1) where k is a positive number.
(a) Calculate the gradient of AB (My answer was -1/7)
(b) Find expressions for the gradient of:
(i) AC - I got (-1-k)/3
(ii) BC - I got k/4
(c)Find the value of k for which angle ACB + 90 degrees
( I know that m1m2 = -1, but I keep getting a negative answer for k)
Your help is very much appreciated! Click here to see answer by venugopalramana(3286)
Question 26141: If each side of a square were increased by 4in., the perimeter would be 8 in. less than twice the perimeter of the orginal square. Find the length of a side of the orginal square.
One squeare has x for orginal and the other x+4 for the new square. Click here to see answer by stanbon(57377)
Question 32573: Hello Tutors,
Here is a question that is making me bang my head as the answr choices don't match my answer. Please explain to me. Thanks
The area of a square region garden is A square feet and the perimeter is P feet. If A = 2P + 9, what is the perimeter of the garden, in feet?
Question 33390: Three angles of a triangle are x degrees and x degrees and x degrees divided by 2.
I need to know if x is greater
or if 60 degrees is greater
or if they are equal
or if there is not enough information to know the answer?
please help! Click here to see answer by Earlsdon(6287)
Question 33427: 1.find the volume of a rectangular solid 15.7 cm by 2 cm by 3cm
2. find the volume of a cylinder whose base has a diameter of 14ft & whose height is 23 ft
3. volume of a square pyramid whose side is 12 cm & whose height is 12cm Click here to see answer by sarah_adam(201)
Question: What is the expression for the gradient of line segment AC?
Answer: (-1-k)/3 | 677.169 | 1 |
To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle v/kr (fig. 14) with the diameters kv, /r at right angles ; the latter is to represent the central meridian. Take koP equal to the co-latitude of the given place, say u; draw the diameter of the poles, through which all the circles representing meridians have to pass.
All their centres then will be in a line smn, which crosses pp at right angles through its middle point m. Now to describe the meridian whose west longitude is w, draw pn making the angle opn, = 90° - 0), then a is the centre of the required circle, whose direction as it passes through p will make an angle opg = w with pp'. The lengths of the several lines are op =tan mu ; W. - cot
?,pc ; oin = cotu ; mu= cosec ac cot w.
Again, for the parallels, take Pb = Pc equal to the co-latitude, say c, of the parallel to be projected ; join vb, vc cutting lr in e, d. Then ed is the diameter of the circle which is the required projection ; its centre is of course the middle point of ed, and the lengths of the lines are ad= tan (u - c); oc = tan 1(u
+ c).
The line sa itself is the projection of a parallel, namely, that of which the co-latitude c = 180° - ac, a parallel which passes through the point of vision.
A very interesting connexion, noted by Professor Cayley, exists between the stereographic projection of the sphere on a meridian plane (i.e., when a point on the equator occupies the centre of the drawing) and the projection on the horizon of any place whatever. The very same circles that represent parallels and meridians in the one case represent them in the other
case also. In fig. 15, abs being a projection in which an equatorial point is in the centre, draw any chord ab perpendicular to the centre meridian cos, and on ab as diameter describe a circle, when the property referred to will be observed. This smaller circle is now the stereographic projection of the sphere on the horizon of some place whose co-latitude we may call u. The radius of the first
circle being unity, let ac= sinx, then by what has been proved above co =sinx cotu= cosx ; therefore a = x, and ac = sin a. Although the meridian circles dividing the 360° at the pole into equal angles must be actually the same in both systems, yet a parallel circle whose co-latitude is c in the direct projection abs belongs in the oblique system to some other co-latitude as c'. To
Question: What is the first step to construct a stereographic projection of the sphere on the horizon of a given place? Answer: Draw the circle v/kr with the diameters kv, /r at right angles. | 677.169 | 1 |
determine the connexion between c and c, consider the point t (not marked), in which one of the parallel circles crosses the line soc. In the direct system, p being the pole, pt =1 - tan i(90° - c) - 1+ cot is and in the oblique, pt = ac (tan mac - tan ,1(2c - c')), which, replacing ac by its value sin it, becomes cos i(gt - c') mu cot me' therefore tanisx= tana is the
required relation.
Notwithstanding the facility of construction, the stereo-graphic projection is not much used in map-making. But it may be made very useful as a means of graphical interpolation for drawing other projections in which points are represented in their true azimuths, but with an arbitrary law of distance, as p =.10). We may thus
avoid the calculation of all the distances and azimuths (with reference to the selected centre point) of the intersections of meridians and parallels. Construct a stereographic projection of the globe on the horizon of the given place ; then on this pro. jection draw concentric circles (according to the stereo-graphic law) representing the loci of points whose distances from the centre are
consecutively 5', 10', 15', 20', kc., up to the required limit, and a system of radial lines at intervals of 5'. Then to construct any other projection, - commence by drawing concentric circles, of which the radii are
previously calculated by the law p=f(u), for the successive values of u, 5', 10°, 15°, 20', &c., up to the limits as before, and a system of
radial lines at intervals of 5°. This being completed, it remains to transfer the points of intersection from the stereographic to the new projection by graphic interpolation.
We now conic to the general case in which the point of vision has any position outside the sphere. Let abed (fig. 17) be the great circle section of the sphere by a plane passing through c, the central point of the portion of surface to be represented, and V the point of vision. Let nj perpendicular to Vc be the plane of representation, join mV
cutting pj in f, then f is the projection of any point rn in the circle abc, and of is the representation
of cm. Let the angle corn= v, Ve=•, Vo=is, ef = p ; then, since of : eV = my : gV, A: sin it P= which gives the law connecting a spherical distance u with its rectilinear representation p. The relative scale at any point in this system of projection is given (keeping to our previously adopted
Question: What is the formula for the relative scale at any point in the system of projection? Answer: The relative scale at any point in this system of projection is given (keeping to our previously adopted formula).
Question: What is the relationship between the distance 't' and the angle 'c' in the direct system of stereographic projection? Answer: In the direct system, pt = 1 - tan i(90° - c) - 1 + cot is. | 677.169 | 1 |
Types of Triangles Study Guide: GED Math (page 3)
Triangles
Triangles are three-sided polygons. The three interior angles of a triangle add up to 180 degrees. Triangles are named by their vertices. The triangle pictured is named ABC because of the vertices A, B, and C, but it could also be named ACB, BCA, BAC, CBA, or CAB. The vertices must be named in order, but can start from any one of the vertices.
If you know the measure of two angles of a triangle, you can find the measure of the third angle by adding the measures of the first two angles and subtracting that sum from 180. The third angle of triangle ABC at left is equal to 180 – (50 + 60) = 180 – 110 = 70 degrees.
The exterior angles of a triangle are the angles that are formed outside the triangle.
Adjacent interior and exterior angles are supplementary. Angle y and the angle that measures 50 degrees are supplementary. Angle z is also supplementary to the angle that measures 50 degrees, because these angles form a line. The measure of angle y is equal to 180 – 50 = 130. Because angle z is also supplementary to the 50-degree angle, angle z also measures 130 degrees. Notice that angles y and z are vertical angles—another reason why these two angles are equal in measure.
The measure of an exterior angle is equal to the sum of the two interior angles to which the exterior angle is not adjacent. You already know angle y measures 130 degrees, because it and angle BAC are supplementary. However, you could also find the measure of angle y by adding the measures of the other two interior angles. Angle ABC, 70, plus angle ACB, 60, is equal to the measure of the exterior angle of BAC: 70 + 60 = 130, the measure of angles y and z.
If you find the measure of one exterior angle at each vertex, the sum of these three exterior angles is 360 degrees. The measure of angle y is 130 degrees. The measure of angle u is 110 degrees, because it is supplementary to the 70-degree angle (180 – 70 = 110) and because the sum of the other interior angles is 110 degrees (50 + 60 = 110). The measure of angle w is 120 degrees, because it is supplementary to the 60-degree angle (180 – 60 = 120) and because the sum of the other interior angles is 120 degrees (70 + 50 = 120). The sum of angles y, u, and w is 130 + 110 + 120 = 360 degrees.
Types of Triangles
Question: How many different names can be given to the triangle ABC? Answer: 6
Question: What is the measure of angle y? Answer: 130 degrees
Question: What is the relationship between adjacent interior and exterior angles? Answer: They are supplementary | 677.169 | 1 |
In geometry, the kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree right triangles with 4, 6, and 12 triangles meeting at each vertex.
Contents
Conway calls it a kisrhombille[1] for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille.
The related rhombille tiling becomes the kisrhombille by subdivding the rhombic faces on it axes into four triangle faces
It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.)
It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.
It is also topologically related to a polyhedra sequence defined by the face configurationV4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
The bisected hexagonal tiling is a useful starting point for making paper models of deltahedra, as each of the equilateral triangles can serve as faces, the edges of which adjoin isosceles triangles that can serve as tabs for gluing the model together.[citation needed]
Question: What is the name of the tiling described in the text? Answer: kisrhombille tiling or 3-6 kisrhombille tiling
Question: What is the label given to this tiling? Answer: V4.6.12 | 677.169 | 1 |
an arrangement of five objects, as trees, in a square or rectangle, one at each corner and one in the middleLagrange (lə-grānj', lə-gränj') Pronunciation Key
Italian-born French mathematician and astronomer who made important contributions to algebra and calculus. His work on celestial mechanics extended scientific understanding of planetary and lunar motion. In 1772 he discovered the points in space that are now named for him.
Question: Which shape is formed by the arrangement of the five objects? Answer: Square or Rectangle | 677.169 | 1 |
In geometry, an inscribedplanarshape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative positionThe shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determinedIn Mathematics, solid geometry was the traditional name for the Geometry of three-dimensional Euclidean space &mdash for practical purposes the kind of Specifically, there must be no object similar to the inscribed object but larger and also enclosed by the outer figure. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other
Familiar examples include circles inscribed in polygons, and triangles or regular polygons inscribed in circles. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called theIn Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are LineGeneral properties These properties apply to both convex and star regular polygons
More precisely, in the phrase "an inscribed F of X", the outer figure X is supposed to be a given, specific figure (such as, for example, "the circle centered at A with radius r"), whereas F stands for a class of figures (such as, for example, "triangle"). Of these figures, an inscribed one is a figure of maximal size among those of the same shape enclosed by X. Usually it is unique in size, but not necessarily in position and orientation.
The definition given above assumes that the objects concerned are embedded in two- or three-dimensionalEuclidean space, but can easily be generalized to higher dimensions and other metric spaces. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within itIn Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
Question: What is the key difference between an inscribed object and a similar object? Answer: An inscribed object is the largest of its kind that can fit inside another object, while a similar object is any object that can be obtained by uniformly scaling another object.
Question: What is the difference between a polygon and a triangle? Answer: A triangle is a polygon with three sides or edges and three corners or vertices, while a polygon can have any number of sides.
Question: What is the minimum number of coordinates needed to specify every point in a two-dimensional Euclidean space? Answer: 2. | 677.169 | 1 |
Proposition 20
In any triangle the sum of any two sides is greater than the remaining one.
Let ABC be a triangle.
I say that in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB.
Similarly we can prove that the sum of AB and BC is also greater than CA, and the sum of BC and CA is greater than AB.
Therefore in any triangle the sum of any two sides is greater than the remaining one.
Q.E.D.
This proposition is known as "the triangle inequality." It is part of the statement that the shortest path between two points is a straight line, but there are many other conceivable paths besides broken lines.
A minimum distance
This proposition on the triangle inequality, along with I.15 on vertical angles, allows us to solve a problem on minimum distance, described and solved by Heron of Alexandria.
Suppose there are two points A and B on the same side of a line CD. The problem is to find the shortest path which goes first from the point A to some point P on the line CD, then from P to the point B. We will only consider paths that are made out of straight lines; call such a path a bent line. But that still leaves us the question of which point P to choose on the line CD to minimize the sum of the distances AP plus PB.
The solution is that the shortest path will be the path AEB where angle of incidence, namely, angle AEC, equals the angle of reflection, namely, angle BED.
First, we should show how to construct the bent line where the angle of incidence equals the angle of reflection.
Draw a perpendicular BF from the point B to the line CD (I.12), and extend it to B' so that FB' = BF (I.Post.2, I.3). Draw AB' and let E be the point where AB' intersects CD. (There will be a point of intersection since A and B' are on opposite sides of CD.) Draw BE.
Now, triangles BFE and B'FE are congruent since they have two sides and the included angle equal (I.4), the included angles being right angles. Therefore, angles BFE and BF'E are equal. The vertical angle AEC across from angle B'ED also equals these angles (I.15). Thus, the angle of incidence AEC equals the angle of reflection BED.
We still have to show that the distance AE + EB is less than any distance AP + EP for any point P other than E that lies on the line CD. Let P be such a point and draw lines AP, BP, and B'P. Then by proposition I.20, above, AP + EP is less than AB'. But AB' = AE + EB', and EB' = EB,
therefore AP + EP is less than AE + EB.
Question: What is the relationship between the sum of any two sides of a triangle and the remaining side, according to Proposition 20? Answer: The sum of any two sides of a triangle is greater than the remaining side.
Question: What are the two angles that are equal in the solution to the problem described by Heron of Alexandria? Answer: The angle of incidence (angle AEC) and the angle of reflection (angle BED). | 677.169 | 1 |
Any right line passing through the center of a figure or body, as a circle, conic section, sphere, cube, etc., and terminated by the opposite boundaries; a straight line which bisects a system of parallel chords drawn in a curve.
The longest distance at right angles, across any circle or cylinder. In standing trees, estimate diameter by dividing the circumference (length of a line taken completely around the outside of a tree) by 3.1416.
(3) A line segment that passes through the center of a circle (or sphere) and has end-points on the circle (or sphere); also, the length of such a line segment. The diameter of a circle is twice its radius.
The diameter of a graph is the length of the longest chain you are forced to use to get from one vertex to another in that graph. You can find the diameter of a graph by finding the distance between every pair of vertices and taking the maximum of those distances.
The length of a straight line passing through the center of a tree or a portion of a tree. Tree diameter is usually measured 4.5 feet above ground level (see DBH), but log diameter is measured at the small end.
The distance of one side of the circle, through its center, and to its other side. The diameter is exactly one half the length of the radius, and there are an infinite number of diameters although they are all the same size because they all must pass through the circle's center. (See Pinhole Camera Observatory)
A straight line segment that passes through the center of a circle or sphere and connects two opposite points. Mathematically, the diameter of a circle is expressed as 2r, with "r" representing the radius of the circle or sphere.
The diameter of a graph is the longest distance you can find between two vertices. When you are measuring distances to determine a graph's diameter, recall that if 2 vertices have many paths of different distances connecting them, you can only count the shortest one. An interesting problem in graph theory is to draw graphs in which both the degrees of the vertices and the diameter of the graph are small. Drawing the largest graphs possible that meet these criteria is an open problem.
A line segment that passes through the center of a circle (or sphere) and has endpoints on the circle (or sphere); also, the length of such a line segment. The diameter of a circle is twice it radius. See also circle.
(n.) The distance across a graph, measured by the number of links traversed. Diameter is usually taken to mean maximum diameter (ie the greatest internode distance in the graph, but it can also mean the average of all internode distances. Diameter is sometimes used as a measure of the goodness of a topology.
Question: What is the diameter of a cylinder? Answer: The longest distance across it at right angles
Question: What is the diameter of a graph in terms of its vertices? Answer: The maximum number of links traversed between any two vertices | 677.169 | 1 |
acute angle traingle - all angles in the traingle has to be less than 90. Two possibilities lets take a or almost a right traingle(89 as one angle -> and the maximum value of the traingle will be somewhat closer to the hypotenue of the traingle.
let x be the other side .. if x is hypotenue it value is less than (12 ^ 2 + 10 ^ 2)^1/2 < 16 .. so max value is 15 let x be the other side .. if 12 is hypotenue, then value of x is greater than (12 ^ 2 - 10 ^ 2)^1/2 > 6 .. so min value is 7
idea behind this ... draw a traingle with 10 as base and 12 as the height and angle formed between these 2 is right traingle. other side is hypotenue and the max value will be 15. then slide the side 12 towards right slowly and at some point side 12 will become hypotense .. and if you move that afterwards towards rights, then angle between 10 and x will be >90Actually, chix475ntu, jeetesh and gurpreet have all given the correct answer either using diagrams or formulas.. 6 < x < 16 is the same as values from 7 to 15 because only integral values are allowed.
Anyway, I am putting down the explanation. The question asks you for an acute triangle i.e. a triangle with all angles less than 90. An obtuse angled triangle has one angle more than 90. So the logic is that before one of the angles reaches 90, find out all the values that x can take. Starting from the first diagram where x is minimum and the angle is very close to 90, to the 2nd diagram where all angles are much less than 90 to the third diagram where the other angle is going towards 90.
Attachment:
Ques1.jpg [ 9.53 KiB | Viewed 1643 times ]
Note: The remaining angle cannot be 90 because that will make 10 the hypotenuse but hypotenuse is always the greatest side.
In the leftmost diagram x = \sqrt{(12^2 - 10^2)} x = root 44 which is 6.something x should be greater than 6.something because the angle cannot be 90.
In the rightmost diagram, x = \sqrt{(12^2 + 10^2)} x = root 244 which is 15.something x should be less than 15.something so that the angle is not 90.Your explanations is easy to assimilate , but i'm having a doubt (may be a silly one) - how do we deicide that x HAS to be the largest side ? Please enlighten [ 17.39 KiB | Viewed 1307 times ] + +
Question: What is the minimum value of 'x'? Answer: 7
Question: Which side is 'x'? Answer: It can be either the side opposite the 10 or the 12, depending on the position of the triangle.
Question: What is the maximum value of 'x'? Answer: 15 | 677.169 | 1 |
Geometry Where can the lines containing the altitudes of an obtuse triangle intersect? I. inside the triangle II. on the triangle III. outside the triangle A)I only B)I or II only C)III only D)I, II, or III PLEASE HELP ME ASAP!!!!
Thursday, December 6, 2012 at 5:09pm by Corey
trig did you make a diagram ? let the top of the monument be A and the bottom be B let the person's position be at P Draw a horizontal from P to AB to meet AB at C. then CB = 9 angle APC = 6°' 50' or 6.8333° angle BPC = 7° 30' or 7.5° Using triangle ...
Tuesday, February 15, 2011 at 12:14am by Reiny
geometry Triangle ABC is congruent to triangle DEF. If the length of AB is 12 feet and the length of AC is 10 feet. Find the length of DE. can someone please help me on this question please and thank youu :)
Saturday, September 3, 2011 at 10:56pm by Alexis
math The scale factor of 2 similar polygons is 3:4. If the perimeter of the large polygon is 200, what is the perimeter of the small polygon?
Wednesday, March 6, 2013 at 9:23pm by sam
geometry The perimeter is the distance around an object. A fence enclosing a backyard marks the perimeter of your backyard.
Wednesday, January 26, 2011 at 9:26pm by Ms. Sue
geometry Suppose the perimeter of a polygon is 27 inches. If the sides of the polygon are scaled by a factor of 1/3, what is the new perimeter?
Tuesday, January 26, 2010 at 10:34pm by sharon
math the scale of two similar quadrilaterals is 1:2. The perimeter of the smaller quadrilateral is 80 centimeters. What is the perimeter of the larger quadrilateral?
Monday, April 5, 2010 at 9:23pm by Anonymous
Math The scale of two similar quadrilaterals is 1:3. The perimeter of the smaller quadrilateral is 90 centimeters. What is the perimeter of the larger quadrilateral?
Wednesday, February 3, 2010 at 9:32pm by Ashely
math Perimeter is 2L + 2W L=3W Perimeter=6W+2W I assume you can take if from here.
Monday, February 9, 2009 at 4:45pm by bobpursley
Question: Where can the lines containing the altitudes of an obtuse triangle intersect? A)I only B)I or II only C)III only D)I, II, or III
Answer: D)I, II, or III
Question: What is the perimeter of a rectangle with length L and width W, given that L = 3W and the perimeter is 6W + 2W?
Answer: The perimeter of the rectangle is 8W. | 677.169 | 1 |
Angles/391965: In a triangle ABC, angle B is 4 times angle A and angle C is 9 degrees less than 5 times angle A.
Find the size of the angles.
A
B
C 1 solutions Answer 278186 by edjones(7569) on 2011-01-08 14:01:42 (Show Source):
Miscellaneous_Word_Problems/391973: Alice and Corinne stand back-to-back. They each take 10 steps in opposite directions away from each other and stop. Alice then turns around, walks toward Corinne, and reaches her in 17 steps. The length of one of Alice's steps is how many times the length of one of Corinne's steps? (All of Alice's steps are the same length and all of Corinne's steps are the same length.) 1 solutions Answer 278183 by edjones(7569) on 2011-01-08 13:42:50 (Show Source):
You can put this solution on YOUR website! When A takes 10 steps back she is where she started. Her 7 steps = C's 10 steps.
the length of A's step is 10/7 that of one of C.
10/7 answer.
.
Ed
Probability-and-statistics/39Linear-systems/391586: below are a rectangle and a right triangle.
The perimeter of the rectangle is 34 cm. The perimeter of the triangle is 30 cm. Find the values of m and n.
On the right triangle, the base is n and the side is m. The slope however is n+1. The rectangles base is n and the side is m. 1 solutions Answer 277855 by edjones(7569) on 2011-01-07 01:40:47 (Show Source):
Permutations/391588: In how many ways can you arrange the letters of the word 'think' so that the 't' and the 'h' are separated by at least one other letter? Use the indirect method. 1 solutions Answer 277853 by edjones(7569) on 2011-01-07 01:24:57 (Show Source):
You can put this solution on YOUR website! Let's make them one letter, the "th" letter. 4!=24
Let's make them one letter, the "ht" letter. 4!=24
24*2=48 ways they are not separated by a letter.
5!=120 possible arrangements of the word "think".
120-48=72 ways you can arrange "think" so the t and h are seperated by at least one letter.
Question: What is the measure of angle A in triangle ABC? Answer: 9 degrees
Question: What is the perimeter of the rectangle in the given problem? Answer: 34 cm | 677.169 | 1 |
Angles
So, what's our angle? We want to help you learn about angles. Really; that's it. Shmoop has distilled our angle knowledge into a short video that will have viewers tossing around words like "acute" and "reflex" in no time. So, if you're looking for a video introduction to the world of angles that is anything but obtuse, you've come to the right place.
Question: What is the tone of the text? Answer: Informative and friendly | 677.169 | 1 |
well an ellipse is an ellipse because its got two foci points and therefore it doesn't have a radius. Whereas a circle only has one focus point and it does have radius where any distance from the focus point to any point in the circumference of the circle would be constant. Now a square is a square because its width and height are equivalent. You can't just say a rectangle is technically a square and an ellipse is technically a circle. They're all different else they wouldn't have different names and mathematical explanations of what they represent.
But other then that... what do you guys use for collisions besides rectangles? or do you use the Geom package at all?
Nobody's saying that squares and rectangles are the same and that ellipses and circles are the same. A circle falls under the category of ellipse, just as a square falls under the category of rectangle. They're both special instances of each category, much like how ellipses and rectangles fall under the category of shape. You could go even more generic with rectangles and say they stem from parallelograms but differ in that their angles all must be right. A line from Wikipedia to clarify the definition of a circle as related to an ellipse is the following:
"A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0."
That's basically a rehashing of what I said previously.
Now, if you want to know how to do collisions with circles.... well, just use the distance formula. It's easier than dealing with ellipses, thanks to the fact that there's only one focus, essentially.
Also circles are so simple I generally wouldn't use the geom versions anyway. Same goes for rects.
So you don't use classes from the geom package for collisions? What do you use?
Correct me if I'm wrong, but a circle is a type of ellipse that just happens to be symmetrical from its center to any point lying on its edge. Also, a square is just a rectangle that has all equal sides. Therefore, it shouldn't matter, should it?
I leaned how to code java with notepad first then I jumped up to notepad++ I was suck a boss. I actually had this guy in my data structures class that was wondering why all his stuff sometimes had to have static in everything and face palmed. He ended up getting a higher grade.
I am not saying static does not matter as it does obviously. Its just that there are many places where use can use it without causing any issues. If I am coding something that will be used by others and needs re-usability, I will do everything properly. For me it seems that when you want really good clean documentation, code, and readability, then you start dropping all the short cuts and hackyness.
Question: What tool did the user start learning Java with? Answer: Notepad | 677.169 | 1 |
Everyone knows what a triangle is, yet very few people appreciate that the common three-sided figure holds many intriguing secrets. For example, if a circle is inscribed in any random triangle, and then three lines are drawn to connect the points at which the circle intersects the triangle with the vertices of the triangle opposite them, these lines will always meet at a common point—no matter the shape of the triangle. Veteran math educators Alfred S. Posamentier and Ingmar Lehmann reveal this and many more interesting geometrical properties in this entertaining and illuminating book about geometry. Flying in the face of the common impression that mathematics is usually dry and intimidating, The Secrets of Triangles proves that this sometimes-daunting, abstract discipline can be both fun and intellectually stimulating.
Authors Posamentier and Lehmann explore the multitude of surprising relationships connected with triangles and show some clever approaches to constructing triangles using a straightedge and a compass. Readers will learn how they can improve their problem-solving skills by performing these triangle constructions. The lines, points, and circles related to triangles harbor countless surprising relationships that are presented here in a very engaging fashion.
Requiring no more than a knowledge of high-school mathematics, and written in clear and accessible language, The Secrets of Triangles will give lay readers new insights into some of the most enjoyable and fascinating aspects of geometry
Question: How does the book help readers improve their problem-solving skills? Answer: By performing triangle constructions, readers can improve their problem-solving skills. | 677.169 | 1 |
In mathematics, the trigonometric functions (also called circular functions)In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations.
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History
The notion that there should be some standard correspondence between the length of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is just these ratios that the trigonometric functions express.
Madhava of Sangamagramma (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.
A few functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin²(θ/2)), the haversine (haversin(θ) = versin(θ) / 2 = sin²(θ/2)), the exsecant (exsec(θ) = sec(θ) − 1) and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). Many more relations between these functions are listed in the article about trigonometric identities.
Right triangle definitions
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.
In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A:
We use the following names for the sides of the triangle:
The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
The opposite side is the side opposite to the angle we are interested in, in this case a.
The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.
Question: What are the six basic trigonometric functions? Answer: Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant (sec), and Cosecant (csc) | 677.169 | 1 |
In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.
The law of cosines is mostly used to determine a side of a triangle if two sides and an angle are known, although in some cases there can be two positive solutions as in the SSA ambiguous case. And can also be used to find the cosine of an angle (and consequently the angle itself) if all the sides are known.
Other useful properties
Periodic functions
Animation of the additive synthesis of a square wave with an increasing number of harmonics
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe the simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of the uniform circular motion.
Trigonometric functions also prove to be useful in the study of general periodic functions. These functions have characteristic wave patterns as graphs, useful for modeling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations. For example the square wave, can be written as the Fourier series
.
In the animation on the right it can be seen that just a few terms already produce a fairly good approximation.
See also
All Students Take Calculus (A mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane)
Sine is a trigonometric function. Sine or Sines may also refer to:
Sines, Portugal, a city and municipality in Portugal
Short interspersed nuclear element, in eukaryote genomes
Sine, Kurdistan, a city in Iranian Kurdistan
.....Click the link for more information.
In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplifiedfunction expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output"). .....Click the link for more information.
angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept .....Click the link for more information.
A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.
In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e. .....Click the link for more information.
Question: What does the law of cosines also allow to find? Answer: The cosine of an angle (and consequently the angle itself) if all the sides are known | 677.169 | 1 |
Trigonometry/Verifying Trigonometric Identities
To verify an identity means to prove that the equation is true by showing that both sides equal one another.
There is no set method that can be applied to verifying identities; there are, however, a few different ways to start based on the identity which is to be verified.
Trigonometric identities are used in both course texts and in real life applications to abbreviate trigonometric expressions. It is important to remember that merely verifying an identity or altering an expression is not an end in itself, but rather that identities are used to simplify expressions according to the task at hand. Trigonometric expressions can always be reduced to sines and cosines, which can be more manageable than other functions.
To verify an identity
Always try to reduce the larger side first.
Sometimes getting all trigonometric functions on one side can help.
Remember to use and manipulate already existing identities. The Pythagorean identities are usually the most useful in simplifying.
Remember to factor if needed.
Whenever there is a squared trigonometric function such as sin2(t), always use the Pythagorean identities, which deal with squared functions.
Sometimes doing the reverse of the normal steps helps. For example, adding 1 in unique forms (such as can help simplify expressions by matching denominators and simplifying numerators.
Question: What is the purpose of adding 1 in unique forms when simplifying expressions? Answer: To help simplify expressions by matching denominators and simplifying numerators. | 677.169 | 1 |
The theorem was generalized by Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points. Then if 2n of those points lie on a common line, the last point will be on that line, too.
Proof of Pascal's theorem
The following proof will actually be just for a single unit circle in the projective plane, but a conic section can be turned into a circle by application of a projective transformation, and since projective transformations preserve incidence properties, then the proof of the circular version should imply the truth of the theorem for ellipses, hyperbolas, and parabolas. Ellipses in particular can be turned into circles by a rescaling of the plane along either the major or minor axis, and a circle of any size can be turned into a unit circle by simultaneous and proportional rescaling of both the x- and y-axes.
Let P1, P2, P3, P4, P5, and P6 be a set of six points on a unit circle of a projective plane, with the following homogeneous coordinates:
P_1 : [cos theta_1 : sin theta_1 : 1]
P_2 : [cos theta_2 : sin theta_2 : 1]
P_3 : [cos theta_3 : sin theta_3 : 1]
P_4 : [cos theta_4 : sin theta_4 : 1]
P_5 : [cos theta_5 : sin theta_5 : 1]
P_6 : [cos theta_6 : sin theta_6 : 1] .
Pascal's theorem then states that the three points which are the intersections of: (1) lines P1P2 and P4P5, (2) lines P2P3 and P5P6, and (3) lines P3P4 and P6P1, are collinear.
Symbolically, this can be stated as:
((P_1 times P_2)times (P_4 times P_5))cdot ((P_2 times P_3)times (P_5 times P_6))times ((P_3 times P_4)times (P_6 times P_1)) = 0,
Question: What kind of curves can Möbius' generalization apply to, apart from circles? Answer: Ellipses, hyperbolas, and parabolas | 677.169 | 1 |
Laura's Answer:
A simpler explanation would be all angles inside a square are equal to 90°. If you have a shape that has 4 equal sides, but only opposite angles are equal, it's a diamond. A diamond is a type of parallelogram.
Laura's Answer:
First off, we need the equations to really help you with this. I can only give you a generic answer without them.
I can tell you that for (a) to be correct, you will get an ordered pair (i.e. one value for x and one value for y). For (b) to be correct, you will get some useless-but-true answer (i.e. 8=8). Usually this means that both equations are different forms of the same line. For (c) to be correct, you will get an untrue answer, such as 0=-4.
this is a stats class. in a sample of all majors at a large university(n=3200), the grade pt average is 2.9 with a standard deviation of 0.9. is there a significant difference between the sample avg and the university(3.1) as a whole at a 92% confidence level. i use the formula 2.9-3.1 over .9/sq rt of 3200.
Laura's Answer:
Hi Mildred,
It sounds like what you are doing in this question is a two-tailed hypothesis test. Also, from what you've given, it seems like you don't have the standard deviation for the whole university (population SD). Therefore, you would normally want a two-tailed t-test, however, since the sample size is so large, you can probably assume a normal distribution and use a Z-test.
First, you want to come up with your null and alternative hypotheses: H0: μ = 3.1 (true mean is 3.1, so sample and population are not different); HA: μ ≠ 3.1 (true mean is not 3.1, so sample and population are different)
To calculate the Z-statistic, use: Z = (X - μ)/(σ/√n)
Where μ = population mean (whole university), X = sample mean, σ = SD and n = number in sample. You have the formula right in your question.
Now, since you are using a 92% CI, but really you have a two-tailed situation, each tail will have 4%. You want to find that 4% on a Z chart (added to 92%). I found that ~.96 occurs at Z ≤ 1.75. So, if you want to reject H0, your Z value must be smaller than -1.75 or larger than 1.75. Remember, H0 is that your sample mean and population mean are the same (i.e. NOT significantly different).
Question: What shape has four equal sides but only opposite angles are equal? Answer: A diamond | 677.169 | 1 |
POLYGON_AREA_2_2D computes the area of a polygon in 2D
(second algorithm);
POLYGON_CENTROID_2D computes the centroid of a polygon in 2D;
POLYGON_CENTROID_2_2D computes the centroid of a polygon in 2D
(a second algorithm);
POLYHEDRON_VOLUME_3D computes the volume of a polyhedron in 3D;
POLYHEDRON_VOLUME_2_3D computes the volume of a polyhedron in 3D
(a second algorithm);
POLYHEDRON_CENTROID_3D computes the centroid of a polyhedron
in 3D;
TETRAHEDRON_CIRCUMCENTER_3D computes the circumcenter of a
tetrahedron in 3D;
TRIANGLE_CIRCUMCENTER_2D computes the circumcenter of a
triangle in 2D;
Question: Which function computes the volume of a polyhedron in 3D? Answer: POLYHEDRONVOLUME3D | 677.169 | 1 |
Archive for December, 2008
Suppose we are given two lines in , and . Line passes through the point and its direction is given by vector
.
Line passes through the point and its direction is given by vector
.
What is the distance between lines and ?
__________
Lines as functions
Let vector-valued functions and be defined as
.
Lines and are thus the images of set under functions and , respectively
.
Function maps each into point . Therefore, choosing a particular value of corresponds to choosing a point on line , and vice versa (because function is injective). Thus, each point on the line can be unambiguously specified by a value of . Similarly, following the same line of thought, each point on the line can be unambiguously specified by a value of .
__________
Distance between two points
In order to compute the distance between lines and , we first need to know how to compute the distance between two points in , i.e., we need to choose a metric on . For the sake of simplicity, we will use the Euclidean metric, i.e., we will use the 2-norm to measure lengths and distances. In other words, we will be working in normed vector space .
Let us fix parameters and , i.e., and , and compute the distance between fixed points and
.
Pictorially, we have that is the length of the line segment connecting fixed points and
To each pair of parameters , it is assigned a non-negative scalar
.
__________
Distance between a point and a line
Suppose now that we vary , while remains fixed, . As we vary , we travel along line and hence the distance between free point and fixed point is given by
.
Since is now free, is a function of .
For example, let us pick three distinct values for , say . The image below depicts the line segments connecting fixed point and points . The length of these line segments gives us the distance between the fixed point and each of the three points on line .
The distance between fixed point and line is thus
.
We have an optimization problem: we would like to find the value of parameter which minimizes function . This minimizer is (see appendix A)
.
Evaluating function at we obtain , the point on line that is closest to fixed point
,
where
is a symmetric, idempotent () projection matrix, and is the orthogonal projection of vector onto vector . It can be shown that vectors and are orthogonal (see appendix B).
For example, let us consider the case: we have two lines in , and , where lies on plane . Pictorially, we have
We would like to find the point on line that is closest to fixed point . Projecting vector orthogonally onto vector , we obtain
The image above illustrates rather nicely that vectors and are orthogonal (see appendix B). The distance between fixed point and line is
,
where . Please note that plane serves no purpose other than to make it easier to visualize in 3-d (plane gives an impression of depth). In other words, plane has some "artistic value", but no "mathematical value".
__________
Question: How does the distance between point P2 and line L1 change as t varies? Answer: The distance is given by ||P2 - (P1 + t*v1)||, which is a function of t.
Question: What is the distance between points P1 and P2? Answer: The distance is not explicitly calculated in the text, but it can be found using the Euclidean distance formula. | 677.169 | 1 |
tangent
cotangent
cosecant
secant
6. There are many trigonometric identities.
The fundamental identities, which are used quite frequently are listed below by
category. It is very important that you know these fundamental identities.
Reciprocal
Quotient
Pythagorean
It
is also helpful to know the algebraic manipulations of some of these
identities.
Ex.:
7. The repetitive behavior of the sine, cosine, and tangent
functions are modeled by the equations in the box at the top of page
448.
8.When using a calculator to find the value of a
trigonometric function, you must be sure that your calculator is in the correct
mode,degrees or radians.
9.
When using a calculator to find the value of the cotangent,
cosecant, or secant of any angle, you must know which trig.
functions are reciprocals of these three. Calculators
do not have keys that will directly evaluate these three functions.
Additional Exercises:
1.
Q: Use even and odd properties to find the exact value of the
expression.
1. In solving certain kinds of problems, sometimes it is
helpful to interpret the six trigonometric functions in right triangles
where angles are limited to acute angles. The inputs
for the functions are the measures of acute angles
in right triangles. The outputs are
the ratios of the lengths of the sides of right
triangles. The ratio of lengths depends on an angle and is a function
of the angle. If the angle is ,
then the six trigonometric functions are functions of . They are defined on page
454. It is extremely important that
you learn these definitions.
2. The
trigonometric function values do not depend on the size
of any triangle. They only depend on the size of the
angle.
3.Within the six trigonometric
functions, there are three sets of reciprocals. It is very
helpful to know which functions are reciprocals of each other.
4.,
, and
are sometimes referred to as special angles. The values of their
trig. functions can be found on page 457.
They can also be derived using formulas for the trig.
functions and the triangles on page 456.
You need to know the exact value of the trig. functions for these
special angles. You need to decide which method of learning them is
best for you. You can memorize the functions for
each angle, or you can memorize the triangles and derive them using the
formulas.
5.
If two angles are complements, the sine of one equals the cosine of the
other. The secant of one equals the cosecant of the other. The
tangent of one equals the cotangent of the other. Because of these
relationships, trigonometric functions can be listed as cofunctions
of each other, and the value of a trigonometric
function of
is equal to the cofunction of the complement
of .
Question: Which of the following is NOT a trigonometric function? A) Tangent B) Cotangent C) Square Root D) Secant Answer: C) Square Root
Question: What are the special angles mentioned in the text? Answer: 0°, 30°, 45°, 60°, and 90° | 677.169 | 1 |
Then you will find solutions for the three questions that another student has posted. First, you will place your solutions to the other student's questions in your course folder. Next, you will provide your solutions to the other student.
When a classmate answers your three questions, let the student know if he or she correctly answered your questions. If the student made a mistake, let this person know where the error occurred and provide the correct detailed answer.
You will be assessed on the following:
the three questions you post, along with the diagram(s) that go with your questions
your answers to a classmate's questions
the feedback you provided to a classmate who has answered your questions (includes accuracy and a detailed solution, as well as politeness)
For example, your project in Google SketchUp may resemble the image below. You could then submit a question and solution, such as the following:
Question 1: If the teepee is 14-ft tall and the teepee makes an angle of 60° with the ground, what is the radius of the teepee?
Solution to question 1: The opposite side is 14 ft. The variable x is the adjacent side.
So when you think of SOH CAH TOA, you can see that you will use TOA or tan.
Lesson 8: The Staircase
Anya has just purchased her first house. It is beautifully finished on the outside.
The inside does need some work, the house is not very large, and there are some space challenges.
Anya has found one problem. The staircase to her upper bedroom is very steep—almost like a slanted ladder. She wants a staircase that is easy to climb. Unfortunately, Anya is unsure of how to design this staircase in the space she has. Can you help?
Your first task is to figure out a new location for the staircase and to sketch a diagram. Since Anya wants a staircase that is easy to climb, you need to look up standard rise and run. You could research using the Internet by entering the keywords "staircase standard rise run."
You will not likely have a diagram detailed enough to answer Anya's questions about your design. Your new knowledge of trigonometry and the Pythagorean theorem will help you answer these questions.
What is the length of the new staircase?
What is the height and width of each new stair?
What is the difference between the angle made by the new staircase and the old staircase?
Question: What are the specific questions Anya has about the new staircase design? Answer: The length of the new staircase, the height and width of each new stair, and the difference between the angle made by the new staircase and the old staircase.
Question: Who should the student provide their solutions to next? Answer: The other student who posted the questions. | 677.169 | 1 |
If two lines intersect, then they intersect in exactly one point.
But this in fact is a Theorem since it follows logically from postulate 1, and we can prove it using the contrapositive of that postulate provided we first translate the postulate to if-then form:
Postulate 1: If A and B are two different points, then there is exactly one line containing them.
Contrapositive: If there are no lines or more than one line containing points A and B, then A and B are the same point.
Thus, if points A and B are contained in both lines l and m, then they must be the same point.
Here is another theorem that seems as simple and obvious as a postulate:
There is exactly one plane containing a given line and point not on the line.
It can be proved directly using postulates 2 and 7:
Proof: By postulate 2, there are 2 points on the given line. Since the given point is not on that line, the given point is a 3rd point, and these three points are noncollinear. By postulate 7, these three points lie in exactly one plane.
These are not all the postulates of Euclidean geometry. Another postulate that Euclid gave (though in a slightly different form) is the so-called Parallel Postulate:
9. Given a line and a point not on that line, there is exactly one line containing the given point and parallel to the given line.
Euclid did not like this postulate because it seemed more complicated than the others, and for more than 1000 years people tried to prove it on the basis of the other postulates. Eventually it was understood that this postulate cannot be proved from the others, and it is possible to assume it false and derive a different kind of geometry (called "non-Euclidean"). In fact, the geometry Einstein used to describe gravitational forces in the universe is a form of non-Euclidean geometry. According to Einstein's Theory of Relativity, all lines in space eventually intersect, so there is no such thing as true parallel lines. But for all practical purposes, Euclidean geometry does work for most of the things we do (as in making buildings) and it is far simpler to use than Einstein's geometry. It is also more intuitive and easier to understand.
Postulates and Theorems Related to Planes
A plane goes on forever, so we can't really draw one. But we can draw a square piece of a plane to illustrate some of the postulates and theorems about planes. Here are the main facts:
(1)
The intersection of two planes is a line:
(2)
Three noncollinear points determine a plane:
(3)
A line and a point not on that line determine a plane:
(4)
Two intersecting lines determine a plane:
(5)
The intersection of a line and a plane not containing that line is a point:
(6)
If a plane contains two points, then it contains the line through those two points.
(7)
It is possible for three or more planes to intersect in a line:
Question: What is the intersection of a line and a plane not containing that line, according to fact (5) about planes? Answer: A point.
Question: How many points are needed to determine a plane, according to fact (2) about planes? Answer: Three noncollinear points.
Question: How many points on a given line are needed to determine a plane, according to Postulate 2? Answer: Two points. | 677.169 | 1 |
This is the basic interface element in both TrigAid and Physics 101 SE. This is called a formula box, it allows the calculation of the main formula, but also its inner variables. Clicking the calculate button yields the main calculation (editfield in gray) and clicking the radial buttons yields the subcalculations. Clicking the "?" shows the formula in either a popup window or below.
The geometry section relies on the formula box for its user interface. Simply enter in your given information into the editfields and click calculate. You may then click the "?" button to see the formula used and a numerical representation of the equation. In addition, clicking the radial buttons (round interface elements) gives the calculatin of that variable when the other fields are supplied with data.
The Triangulator is a powerful feature, it will try to find based upon the information given to it. First ascertain the type of triangle it is and select it in the popupmenu. If any of the sides is given by an exact value, click the appropiate checkbox to use root notation. Next enter in as much information as you can about the sides and angles and click calculate. The program will try to fill in as much information as it can.
This is a straightforward section. In the first two altered formula boxes is the law of sines and cosines. You must select through a popupmenu the side or angle you are solving for, enter in the required information and click calculate.
The Famous SOHCAHTOA is set up according to the foundation of its saying so it can be easily used. Just enter in the information and click on the radial button of the unknown quantity.
Lastly there is Pythagoren's Theorem, enter in two sides and click the radial button of the desired unknown quantity. Finally, you can find the formula by clicking "?" or the buttons in the SOHCAHTOA groupbox.
To find the zero of a polynomial enter in the coefficient into the equation above, put a zero in a term that does not exist. Click the "Insert Coefficients" button and uses the up and down arrows to find a zero of that polynomial. A zero is found when the last position is zero. When this is found click "Shift and Store" and repeat the process as needed.
The first formula box converts a log based expression into one including an exponent. The second uses the change of base formula to calculate logs with uncommon bases. The third one solves for a exponent and creates a logarithm and finally the last solves exponential growth and decay problems.
Enter in the magnitude (a length, or velocity, etc) and their corresponding angle in degrees. This will yield a vertical magnitude, horizontal magnitude, overall magnitude and angle. It is important that the magnitudes and angles be entered in the correct pairs, or they will mismatch and produce incorrect answers. The reference angle is there to provide a reference where the angle can be based off of.
Question: How can you view the formula used for a calculation in the geometry section? Answer: Clicking the "?" button shows the formula used and a numerical representation of the equation.
Question: What is the purpose of the 'Insert Coefficients' button in the polynomial zero-finding section? Answer: To insert the coefficients of the polynomial into the equation.
Question: What is the first step in using the Triangulator feature? Answer: First, ascertain the type of triangle it is and select it in the popupmenu. | 677.169 | 1 |
In 1 hour, P travels a distance of 10 kilometers at a bearing of 30 degrees and Q travels a distance of 12 kilometers at a bearing of 300 degrees (-60 degrees).
In 2 hours, P travels a distance of 20 kilometers at a bearing of 30 degrees and Q travels a distance of 24 kilometers as a bearing of 300 degrees.
The triangle formed for the 2 hour distances is still a right triangle labeled ABC where A is the origin of both boats, and B is the point that P has reached, and C is the point that Q has reached. Triangle ABC is therefore labeled with a 90 degree at angle A, and side AB = 20 kilometers, and side AC = 24 kilometers.
Since side AB is opposite angle C, then side AB is called side c.
Since side AC is opposite angle B, then side AC is called side b.
The side opposite angle A is called side a which is formed by the line segment BC.
You want to find the length of side a because it is the distance between the 2 boats.
Using the right triangle formula and since side a is the hypotenuse of the right triangle formed, you get a^2 = b^2 + c^2 which becomes a^2 = 20^2 + 24^2 = 31.2409987 kilometers.
Question: Which side of the triangle is side a? Answer: The line segment BC
Question: Which triangle is formed by the 2-hour distances of P and Q? Answer: Triangle ABC | 677.169 | 1 |
I'm sorry! I hope this helps. (I don't know how to get my attachment any bigger) but I'll try this.
Answers
What i need to find is arc HF where GH = 6 units and IG = 12 units and angle IGA is conguent to angle ACE; to make my circle a little more clear the middle letter is A; vertical cords are IF and BE. Radii is AH and AD. angle G and C are right angles. In addition to that I'll try one more attachment. thanks for trying to help!
Question: What are the radii of the circle? Answer: AH and AD | 677.169 | 1 |
large enough, we can use the shadow cast by a person. This makes it very different from the traditional sundial we see often in parks and gardens where the shadow is cast by a triangular shaped wedge. The analemmatic sundial is perfect as a piece of large mathematical sculpture.
Traditional Dial
Analemmatic Dial
Shape
Hour lines radiate from a central point.
Ellipse of hour points.
Shadow-casting object
Fixed, parallel to Earth's axis.
Changes daily, vertical.
What is the declination of the sun?
Before we can construct our dial we need to understand what the "declination of the sun" is. Look at figure 1, showing the Earth as seen from the side. Imagine you are standing on the surface of the Earth at the point . Your latitude, i.e. your angular distance from the equator, is marked as . The diagram shows the situation at Noon so the sun is at its high point for the day and will appear to be due South from where you stand.
Figure 1: Calculating the declination of the sun.
Exercise 1: Using this diagram, show that the angle that the sun makes with the horizontal at Noon is
(1)
Calculate this for your own latitude at mid-summer and mid-winter.
On any given day the declination of the sun, marked as in figure 1, is the angle the rays of the sun make with the plane of the equator of the Earth. Because the axis of the Earth is tilted and we rotate around the sun, this angle changes during the year. It is this change that gives us the seasons.
In the Northern hemisphere, is positive in summer, and at the spring and autumn equinoxes the sun is directly overhead at the equator at Noon so . At the summer solstice , and at the winter solstice . As the latitude of the equator is the formula above gives . A graph of average values is shown in figure 2 and the average values for the first day of each month are in table 2.
Figure 2: The declination of the sun.
Table 2: Average values of the declination of the sun.
The path that the sun appears to trace in the sky during the day is a circle. Of course part of this circle will be below the horizon during the night! The centre of this circle is a point due North at an angle to the horizontal.
Figure 3: The path of the sun.
This point is marked (to a very good approximation) by the pole star Polaris. The angle between the sun and Polaris is simply .
In a normal sundial, a shadow is cast by a "gnomon" or pointer that points directly towards Polaris. If a disc of card is stuck to this pointer, perpendicular to it, then the shadow cast by the sun is of constant length throughout the day and moves around the card clockwise with the end point of the shadow lying on a circle. Such a sundial (called an equatorial sundial) is
Question: How does the shadow-casting object differ between a traditional sundial and an analemmatic sundial? Answer: In a traditional sundial, it is fixed and parallel to Earth's axis, while in an analemmatic sundial, it changes daily and is vertical
Question: What is the declination of the sun at the spring and autumn equinoxes? Answer: 0 degrees
Question: What is the shape of the hour lines on a traditional sundial? Answer: They radiate from a central point
Question: What is the angle between the sun and Polaris? Answer: δ | 677.169 | 1 |
Basic Geometry Vocabulary Card exciting UNO-like games will have your students mastering basic geometry vocabulary in no time! There are three separate games in this file; Angles, Lines, and Geo Vocab. Some of the vocabulary reinforced includes acute angle, straight angle, line, ray, line segment, parallel lines, and perpendicular lines. For best results, print in color on heavy paper (or on plain paper and glue to heavier paper) and laminate.
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Spreadsheet (Excel) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
130.5
Question: What are some of the geometry terms reinforced in the games? Answer: Acute angle, straight angle, line, ray, line segment, parallel lines, and perpendicular lines. | 677.169 | 1 |
Dividing by 180°, we find 1 - 2/N < 2/M, or NM - 2M -2N < 0. Adding 4 to each side, NM - 2M - 2N + 4 < 4. Factoring this, we find (N - 2)(M - 2) < 4, the condition that must be satisfied by any M, N. We see at once that neither M nor N can exceed 5, so the only possible values are 3, 4, and 5. The following table exhausts all the possibilities:
M
N
Figure
5
3
Icosahedron
3
5
Dodecahedron
4
3
Octahedron
3
4
Cube
3
3
Tetrahedron
The faces of the polyhedron can be only triangles, squares and pentagons. If we try M = 4 and N = 4, we see that four squares can meet at a vertex only if they lie in a plane, and so do not make a polyhedron. Of course, here (4 - 2)(4 - 2) = 2 × 2 = 4, which is not less than 4. M = 3 and N = 6 lead to a similar conclusion. In each case, we have a tesselation of the plane, but no polyhedron. Now it is quite clear why there are only five regular solids, and we know it not because someone has said so, but through our knowledge of space.
That there are only five regular polyhedra has been known for more than 2500 years, but the fact is much less important than the reasoning used to establish it. This method was unique to the classical world of Greece and Rome, and has led to modern science and understanding. It is, unfortunately, still the possession of only a few people, and is sadly lacking in modern mass education.
The illustration at the link to this page is an anaglyph of a dodecahedron. View it through red-green 3-D glasses to see the polyhedron in space.
Question: What is the significance of the reasoning used to establish the number of regular polyhedra? Answer: It is unique to the classical world of Greece and Rome and has led to modern science and understanding
Question: What is the maximum value that M or N can have according to the given condition? Answer: 5 | 677.169 | 1 |
Moving Circles
Hi!
I stumbled upon a problem that I was not able to solve at school with help of my teachers.
Info: There are 2 circles. Each has its own radius and speed. Circle 1 is going from point A to point B. Circle 2 from C to D. They travel in 2D space in any direction and can have the same start, end or both.
All of the above data is known.
Question: When will center of circle 2 be inside circle 1 and when it will go out of circle 1, if it will even be inside? How long is it inside? ( exitTime - entryTime)
Programming language that I'll be using for a solution doesn't really matter ( C++ most likely, but C# is considered as another option). Simulation is not an option.
Re: Moving Circles
Hi;
What are the speeds of the circles? What are the sizes of the circles? How big is the space they are moving in? What law governs their motion? There are many moreI need just basic equation how to calculate this. You can take sample data any you like ( or I can make some up, I don't have any with known solution). Their motion is linear (I hope used a correct phrase).
Re: Moving Circles
Hi;
There is a method to find whether two circles overlapbobbym wrote:
Hi;
There is a method to find whether two circles overlap.
This is a friend of him speaking and I'll try make it as clear as possible.
Imagine you have two tanks that have different ranges of fire (we're trying to make a game :p). These tanks will move in a linear path, the path can be just about anything you imagine. The radius of the circle can also be "random". These paths can intersect, go parellel or move away from each other. As pointed, the range and the size of the circle is also almost any various number.
Let's take an example: The tanks intersect at a 45 degree angle towards each other in a 2d plane of a limited size (this size limit can be made up if you so desire). Tank A: Moves at 20 m/s. Has a range of 50m. Moves for 100m.
Tank B: Moves at 50 m/s. Has a range of 30m. Moves for 200m.
Also, as you can imagine, they have a starting point and an ending point, so it's not always necessary they start in the corner of the plane. They can be at any given point in this limited area.
Now, we have to know for how long the tank A will be shooting at tank B and the other way around. Keep in mind that there can be as many units on the field, but we're using two as an example. What we basically want to know, how to calculate when and for how long the tank will shoot other units.
Hopefully I've made it clear, thank you for taking your time!
Question: What is the range (radius) of Tank B's fire? Answer: 30m
Question: What is the speed of Tank B? Answer: 50 m/s
Question: How far does Tank A travel? Answer: 100m
Question: What is the speed of Tank A? Answer: 20 m/s | 677.169 | 1 |
Chapter Seven
For Septisection
If the Arc BCDEFGHI shall be cut in these seven equal parts with the lines BE, BG, BI being drawn, OD will be 1 , and BO, 2 - 1. And the Subtended Chord [BI] of seven times the Arc [BC] will be, as here concluded, equal to 7 - 14 + 7 - 1 .
With being given therefore eight lines in continued proportion of which the first is the Radius, the second whatever Chord you wish: 7 of the Chord being increased by adding 7, will be equal to 14 + 1 and [i.e. added to] the Subtended Arc of seven times [the angle].
And by this equation: with any given Subtended [Chord], we will be able to find the Subtended Arc of seven times; or if seven of the given Arc shall be in excess of one, two or three circles, we will be able to find the Subtended Chord excess above the whole number of circles.
It should be noted with all the equations of the section by this method, if the number of the entire Circles shall be less than one: the same signs of the equation will be used, because the Periphery together come to less than 360 Degrees; if on the contrary it will have been more than a whole circle, the opposite signs should be placed. Let the given Subtended Chord [angle] be 140 Degrees. Seven times 140 has the value 980. That is 260 beyond two whole circles.2
Let the Subtended Chord be 157:2':40": 196000039119128.
Seven times 157:2':40" has the value 1099:18':40" or 19: 18':40" beyond three whole circles.
Because the number of Circles is more than a whole circle the equation will be
14 - 7 - 7 + 1 . [See Note 2]
Let the given Subtended Chord of 10 Degrees be 17431148549. The Chord of 70:0' being sought:
2. We are able also by the same equation: For any given Subtended Arc, to find the Subtended Chord of the seventh part of the same Arc, or the total being composed from the given Arc, and with one, two, or three circles.
The way of working will be scarcely different from that which was expounded above, for the Subtended Chords of the Third and Fifth parts. But the working for the multiplication of the terms is not without a little more labour.
Question: What is the equation to find the Subtended Arc of seven times with a given Subtended Chord? Answer: With any given Subtended Chord, we will be able to find the Subtended Arc of seven times
Question: What is the length of BO in the same arc? Answer: 2 - 1
Question: What is the equation if the number of entire Circles is less than one? Answer: The same signs of the equation will be used | 677.169 | 1 |
And by this easiest of methods the equations are being found and are able to be demonstrated, if any periphery you wish being cut into equal parts however big, by the method shall be the number deficient [determined]: which all will give the Subtended Chords of multiplied Arcs, by the same preceding method, and the parts also, but not with the same facility. For where there are many Arcs of equal segments, with these there are equations with more terms, and to any term being added a larger number: As with all operations the other will be the source of more difficulty.
Notes On Chapter Seven
1 This beautiful piece of elementary geometry is based on the set of similar isosceles that Briggs considered previously; see the Notes 1 of Chapters Three and Five. The constructions 'work' because the angle subtended at the centre is double the angle subtended at the circumference by the equal angles GBI, GBE, and EBC. As AH bisects the arc GI, etc, it follows that the angles at the centre such as HAI are equal to the angles GBI, etc. The rest follows from the set of similar isosceles discussed previously. Thus, for the first set of proportionalities, [Table 7-1] becomes:
Question: What is the method used to find the number of deficient parts in a periphery? Answer: The method of cutting the periphery into equal parts | 677.169 | 1 |
Post a reply
Topic review (newest first)
MathsIsFun
2005-10-11 07:31:25
Likewise I get 9 for the first and 8 for the second. Maybe you were supposed to add some lines?
How many in this one:
mathsyperson
2005-10-11 03:32:43
I can only find 9 in the first one and 8 in the second one. Maybe one of the other members can offer some dramatic revelation that makes the rest if us look stupid, but if not then your teacher's probably just made a mistake.
kuba.g
2005-10-11 01:43:25
Hi,
Our maths teacher gave us this puzzle:
He first showed us the triagle above and asked how many triangles are in it. Everyone said 9. But he said it there ware 35, and he asked if we can find the formula for it.
He also showed this: This one is ment to be 25 triangles???
Does anyone know the formula for it? Please don't use too complicated terms as I'm only 13 It would be nice if I'm the only one of the class to know the answer.
Question: What did mathsyperson find in the first and second triangles? Answer: They found 9 triangles in the first one and 8 in the second one. | 677.169 | 1 |
With this song, as is true with most of the others, students need clear
instruction that includes manipulatives and plenty of practice. I have found
that the basic concept of, "What is an angle?" can be very confusing for
some students.
I believe that the hand motions on this song are absolutely essential. After
all, students are singing, "Square corner just like so." If students don't
show the angle with their hands, then what do they mean when they sing,
"...just like so?" Watch a few students
in my class doing the hand motions to this song.
Question: If students don't use hand motions, what does it imply about their understanding of the line "Square corner just like so"? Answer: It implies that they might not understand what they mean when they sing that line. | 677.169 | 1 |
Question 303246: your boat is traveling due north at 20 miles per hour. a friend's boat left at the same time and is traveling due west at 15 miles per hour. after an hour you get a call from your friend who tells you that he has just stopped because of engine trouble. how far must you travel to reach your friend? Click here to see answer by [email protected](15645)
Question 303518: my algebra 1 class is doing pythagorean theorum, and I got stuck on the very last part of it, which should be easy but since it is not a perfect square I forgot how to do it. c squared = 89. how do i get an answer not in decimal or fraction form? Click here to see answer by jim_thompson5910(28536)
Question 310470: 1.Determine the distance between A(-1, 2, 3) and B(4, -5, 6) to the nearest tenth.
2.Which of the following is the equation in slope-intercept form of the line that passes through (0, 2) and has a slope of -3.
a)y = -3x + 2
b)y = 2x - 3
c)2/3x+0
d)-3/2+0
3.Determine the coordinates of the midpoint of the line segment whose endpoints are A(0, 2, 4) and B(-8, 2, -6).
4.Find the coordinate of the image of point (-1, 3) reflected over the x-axis.
5.What is the measure of the angle of rotation with respect to two lines that intersect to form a 25° angle?
Question 311667: Not sure how to explain these but here goes...
The directions are Use the Pythagorean Theorem to find the missing side. Then calculate the perimeter.
This is a trapezoid with a right angle. The top base is 30. The bottom base is 60 and the leg of the right angle is 20. I need to find X but for the life of me I do not know how. Thanks for your help if you understand.
Then this one.
Another trapezoid. Top base is 30 and bottom base is 58. The height is 50. Need to find the two legs which are each X.
Question 311844: In the right triangle, find the length of the side not given. Give an exact answer and an approximation to three decimal places. a=5, b=7
what is the exact value of c?
what is the value of c approximated to 3 decimal places? Click here to see answer by nerdybill(6951)
Question 311874: The directions are Use the Pythagorean Theorem to find the missing side. Then calculate the perimeter.
Question: In the algebra problem, what is the value of c when expressed as a mixed number? Answer: The value of c is 9 + √89/2, which can be approximated as 9 + 4.71699/2, or 9 + 2.358495, which is approximately 11.358495.
Question: In the algebra problem, if c² = 89, what is the value of c when rounded to the nearest whole number? Answer: The value of c is 10 when rounded to the nearest whole number.
Question: In the algebra problem, what is the square root of 89? Answer: The square root of 89 is approximately 9.433981132.
Question: In the first scenario, if you had been traveling for 2 hours instead of 1, how much further would you be from your friend? Answer: You would be 20 miles further, as you travel at 20 miles per hour and your friend's boat is stationary due to engine trouble. | 677.169 | 1 |
Vector MultiplicationThere are two forms of vector multiplication: one results in a scalar, and one results in a vector.Dot ProductThe dot product , also called the scalar product, takes two vectors, "multiplies" them together, and producesa scalar. The smaller the angle between the two vectors, the greater their dot product will be. A commonexample of the dot product in action is the formula for work, which you will encounter in Chapter 4. Work is ascalar quantity, but it is measured by the magnitude of force and displacement, both vector quantities, and thedegree to which the force and displacement are parallel to one another.The dot product of any two vectors, A and B , is expressed by the equation:where is the angle made by A and B when they are placed tail to tail.The dot product of A and B is the value you would get by multiplying the magnitude of A by the magnitude ofthe component of B that runs parallel to A . Looking at the figure above, you can get A B by multiplying themagnitude of A by the magnitude of , which equals . You would get the same result if you multipliedthe magnitude of B by the magnitude of , which equals .Note that the dot product of two identical vectors is their magnitude squared, and that the dot product of twoperpendicular vectors is zero.EXAMPLESuppose the hands on a clock are vectors, where the hour hand has a length of 2 and the minute hand has a length of4. What is the dot product of these two vectors when the clock reads 2 o'clock?The angle between the hour hand and the minute hand at 2 o'clock is 60. With this information, we cansimply plug the numbers we have into the formula for the dot product:
Question: In the given example, what is the length of the minute hand vector? Answer: The length of the minute hand vector is 4.
Question: What is the dot product also known as? Answer: The dot product is also known as the scalar product. | 677.169 | 1 |
Explanations1. ABy adding A to B using the tip-to-tail method, we can see that (A) is the correct answer.2. AThe vector 2A has a magnitude of 1 0 in the leftward direction. Subtracting B , a vector of magnitude 2 inthe rightward direction, is the same as adding a vector of magnitude 2 in the leftward direction. Theresultant vector, then, has a magnitude of 10 + 2 =12 in the leftward direction.3. DTo subtract one vector from another, we can subtract each component individually. Subtracting the x -components of the two vectors, we get 3 –( –1) = 4 , and subtracting the y -components of the two vectors,we get 6 – 5 = 1 . The resultant vector therefore has an x -component of 4 and a y -component of 1 , so thatif its tail is at the origin of the xy -axis, its tip would be at ( 4 , 1 ) .4. DThe dot product of A and B is given by the formula A B = A B cos . This increases as either A or Bincreases. However, cos = 0 when = 90, so this is not a way to maximize the dot product. Rather, tomaximize A B one should set to 0 so cos = 1.5. DLet's take a look at each answer choice in turn. Using the right-hand rule, we find that is indeed avector that points into the page. We know that the magnitude of is , where is the anglebetween the two vectors. Since A B = 12 , and since sin , we know that cannot possibly begreater than 1 2 . As a cross product vector, is perpendicular to both A and B . This means that it hasno component in the plane of the page. It also means that both A and B are at right angles with the crossproduct vector, so neither angle is greater than or less than the other. Last, is a vector of the samemagnitude as , but it points in the opposite direction. By negating , we get a vector that isidentical to
KinematicsKINEMATICS DERIVES ITS NAME FROM the Greek word for "motion," kinema . Before we can make anyheadway in physics, we have to be able to describe how bodies move. Kinematics provides us with thelanguage and the mathematical tools to describe motion, whether the motion of a charging pachyderm or acharged particle. As such, it provides a foundation that will help us in all areas of physics. Kinematics is mostintimately connected with dynamics: while kinematics describes motion, dynamics explains the causes for thismotion.
Question: What is the name of the branch of physics that deals with the description of motion? Answer: Kinematics
Question: What does the dot product of two vectors represent? Answer: The dot product of two vectors represents the product of their magnitudes and the cosine of the angle between them.
Question: What is the magnitude of the vector 2A in the leftward direction? Answer: The magnitude of the vector 2A in the leftward direction is 10. | 677.169 | 1 |
Standard angles that share the same terminal side are called coterminal angles. They differ by an integer number of full revolutions counterclockwise or clockwise.If the angle is measured in radians, then its coterminal angles are of the form:(theta) + 2(pi)n, where n is any integer.If the angle is measured in degrees, then its coterminal angles are of the form:(theta) + 360ndegrees, where n is any integer.
How to define trig functions
SOH-CAH-TOASin(theta)=Opp/HypCos(theta)=Adj/HypTan(theta)=Opp/AdjWe can define more trig functions by taking the reciprocal of these. We end up with Csc, Sec, and Cot.Also,tan(theta)=sin/cos=rise/run=slope of terminal side.
"Brothers"
Brothers are angles that have the same reference angle.For example, the angles of 30, 150, 210, and 330 degrees are brothers; they all have the same reference angle of 30 degrees, or (Pi/6) radians.Famous Positive Brothers(Pi/6), (Pi/4), and (Pi/3)Again, for (Pi/6)...look at the patternQ1: (Pi/6)Q2: (5Pi/6); observe that 5 is 1 less than 6Q3: (7Pi/6); observe that 7 is 1 more than 6Q4: (11Pi/6); observe that 11 is 1 less than twice 6This pattern follows the others:Q1: (Pi/4)Q2: (3Pi/4)Q3: (5Pi/4)Q4: (7Pi/4)and Q1: (Pi/3)Q2: (2Pi/3)Q3: (4Pi/3)Q4: (5Pi/3)
How do signs of trig values differ between quadrants?
Remember: "All Students Take Calculus"* All of the six basic trig functions are positive in Q1* Sin and its reciprocal, Csc, are positive in Q2* Tan and its reciprocal, Cot, are positive in Q3* Cos and its reciprocal, Sec, are positive in Q4
Among the six basic trig functions, cos (and its reciprocal, sec) are even:cos(-x) = cosxsec(-x) = secx, when both sides are definedHowever, the other four (sin and csc, tan and cot) are odd:sin (-x) = -sinx csc (-x) = -cscx, when both sides are definedtan (-x) = -tanx, when both sides are definedcot (-x) = -cotx, when both sides are definedNote: If f is an even function (such as cos), then the graph of y=f(x) is symmetric about the y-axis.Note: If f is an odd function (such as sin), then the graph of y=f(x) is symmetric about the origin.
Question: What is the sign of sin(-x)?
Answer: sin(-x) = -sin(x) | 677.169 | 1 |
Prove that tangents to a circle at the endpoints of a diameter are parallel. Thank you!
Answers
Lines that are tangent to a circle are perpendicular to the radius at the point of tangency. A diameter is formed by two radii that go in opposite directions from the center of the circle. It is a straight line because the angle would be 180 degrees. Since the tangent lines are perpendicular, the angles between the radii and tangent lines are 90 degree. Since the diameter is a straight line, it is now a transversal line cutting across the two tangents. The alternate interior angles are equal, they are both 90 degrees. When a transversal cuts two lines and forms equal alternate interior angles, the two lines are parallel.
Question: What is the angle formed by a diameter and the radii of a circle? Answer: 90 degrees | 677.169 | 1 |
From Higher Dimensions Database
A simplex is an n-dimensional polytope with n+1 facets and n+1 vertices. Each simplex's element counts, treated as a list and including a single "-1D element" and a single element of its own dimension, read identical to a row of Pascal's triangle.
Simplices are special because they are always convex and are never self-intersecting. Any polytope can be defined as the union of a set of simplices of its own dimension, or as the space bounded by the union of a set of simplices of one dimension less. For these reasons, simplices are often used as "building blocks" in CGI.
Regular simplices are all also self-duals. The only other regular polytope to exhibit this behavior, bar 2D shapes, is the icositetrachoron in 4D.
Under the Tamfang naming scheme, simplices are denoted by the pyro- prefix, meaning the classical element of "fire".
Question: What is the relationship between the element counts of a simplex and Pascal's triangle? Answer: Each simplex's element counts, treated as a list, including a single "-1D element" and a single element of its own dimension, read identical to a row of Pascal's triangle. | 677.169 | 1 |
Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number.
F-TF.4
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F-TF.6
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
F-TF.7
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
F-TF
Prove and apply trigonometric identities
F-TF.8
Prove the Pythagorean identity sin2(theta) + cos2(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.
F-TF.9
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
G
Geometry
G-CO
Congruence
G-CO
Experiment with transformations in the plane
G-CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.2G-CO.5G-CO.6
G-SRT.2
Given
G-C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
G-GPE.5
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
S-ID.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
S-ID
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S-ID.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6.a
Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
S-ID.6.b
Question: What is the purpose of using the unit circle in trigonometry? Answer: To express trigonometric values for angles in terms of their values for any real number x.
Question: What is the relationship between the length of an arc and the radius of a circle, as derived using similarity? Answer: The length of the arc is proportional to the radius. | 677.169 | 1 |
So, in any real road you're going slightly sideways. The road is not built flat ON PURPOSE. It has the shape of a roof. That is called "pumping" in Spanish (bombeo) and "crown" in English: the road is inclined, 2% or so, sideways, for water to leave the road when it rains.
Now, you know (if you didn't knew already) that a road is made of straights that have a slight lateral inclination. Moreover, in most modern circuits (but not all, Catalunya is a typical example) you have:
1. straights 2. logarithmic spirals (that is, curves that go from zero curvature to a definite one, like the trajectory of a coin that you roll on the floor) and 3. circular curves.
Having said all this you have to ask yourself: which is the LATERAL PROFILE of such a road? I mean, if you take a knife and cut the road lengthwise, what do you see?
1. In straights, you see the center axis and the two edges below it. 2. In logarithmic spirals you get a transition of the superelevation. You rotate the road around its axis to change the sideslope from a crown to a full superelevation at the entrance of the circular curve. So, one edge rises and the other sinks. 3. In circular curves you have a constant superelevation: the outer edge is over the axis and the inner edge is below it at a constant height along the curve.
So, a superelevation diagram of a road with spirals is something like this:
The green line is the axis on the straight viewed from the side, the yellow line is the spiral axis and the red line is the circular curve axis.
The dotted lines are the Edges of Traffic Lanes (ETL). On the straight (Points On Tangent or POTs)both edges are at the same height, hence you see only one dotted line. Where the Transition Spiral (TS) begins, the outer edge is flat (at the same height as the axis) but the inner edge have kept its two percent inclination. Where both edges have reached two percent is the Point On Spiral (POS). Where both edges have reached full superelevation is the Point on Curve (POC).
Question: In a circular curve, where is the outer edge in relation to the axis? Answer: Above it. | 677.169 | 1 |
The angle between the surfaces is equal to the angle between the normal-vectors to these surfaces... which you can determine using the dot-product.
You have enough information to determine each normal-vector.
Finding the angle between surfaces
Quote by l46kok
Wait, but I thought normal vector was found using the cross product, not the dot product?
That's one way to do it (and that's the best way for your problem).
Now, once you found those normal vectors using the cross-product,
now use the dot product to find the angle between those [normal] vectors.
If two vectors lie IN a surface, then the normal to the surface must be normal to both. In fact, unless the two vectors lie along the same line (in which case their cross product would be 0) their cross product IS normal to the surface.
The FIRST thing you had better do is set up a coordinate system so that vectors from one point to another make sense! Set up a coordinate system with origin at F, x-axis along FG, y-axis along FE, z-axis along FB. Then the vector perpendicular to CGHD is <1,0, 0> and the vector perpendicular to plane PQR is the cross product of QP= <1,0,1>-<2,1,1>= <-1, -1, 0> and QR= <1,0,1>- <2,0,0>= <-1,0,1>. That cross product is <-1,1,-1>. The angle is given by <1, 0, 0>.<-1, 1, -1>= |<1,0,0>||<1,1,1>|cos([itex]\theta[/itex] or -1= (1)([itex]\sqrt{3}[/itex])cos([itex]\theta[/itex])which gives cos([itex]\theta)= -1/\sqrt{3}[/itex]. What is [itex]\theta[/itex]?
Question: What is the vector perpendicular to CGHD in the given coordinate system? Answer: The vector <1,0,0> is perpendicular to CGHD. | 677.169 | 1 |
It is very useful to be able to create three-dimensional drawings
in Geometry. Many geometric figures are three-dimensional, especially
when you are studying volume and surface area of cubes, pyramids,
cones and spheres.
There are a number of different ways to draw three-dimensional
objects. Examples of a cube drawn using four different types of
drawings are named and illustrated below:
Of the four types of 3D drawing shown above, the best choice for
drawing geometric solids is the Isometric, as it shows the solid
fairly realistically and is easy to draw. You will see an Isometric
Grid below, which you can print and use for your own drawing
projects. Place a piece of tracing paper over the grid, and you can
then experiment with drawing some 3D figures yourself.
The drawing below is an example of a brick (or box) drawn using an
isometric grid. You can practice your 3D drawing-by-drawing simple
solids, such as this one.
Another interesting 3D project is one I call "How Many
Planes. This is a good introduction to three-dimensional
visualization and drawing for students who may never have done any of
this type of drawing.
Some students have trouble with 3-dimensional
geometry. In this worksheet, they will gain expertise in interpreting
a 3-dimensional diagram, and visualizing (as well as drawing) planes.
The more that students see and draw in 3 dimensions, the more
comfortable they will be with this concept.
The ability to understand, and to draw
3-dimensional diagrams will be of great benefit to the students when
they are studying volume and surface area of prisms, pyramids,
cylinders and cones.
The project begins with a question: How many
planes are determined by the vertices of a cube? Use the drawing
below to help you find the answer. Find as many planes as you can.
Shade each plane in on each one of the cubes below. Group them in
categories and label each category at the top of its row. There are
more cubes drawn than you will need.
The answer to this question can be found at the bottom of this
web page.
The drawing below shows a geometric castle. The main part of the
castle is a rectangular prism, and the four towers are, starting with
the front left: a square-based prism with a pyramid on top, a
cylinder with a hemisphere on top, a triangular prism with a pyramid
on top, and a cylinder with a cone on top. The smaller drawing below
shows you the castle from above. Now, my challenge to you is to use
the formulas for volume and surface area to find the total volume and
surface area of this castle! I would suggest you do each geometric
solid separately (of course), show all your work, and keep them in an
organized fashion on your paper. You will find the answers at the
bottom of this web page - but don't look until you are finished!
Question: What is the name of the 3D drawing project that helps students understand and visualize planes? Answer: "How Many Planes"
Question: What is the first step in the "How Many Planes" project? Answer: To find out how many planes are determined by the vertices of a cube | 677.169 | 1 |
How to Recognize Basic Trig Graphs
The graphs of the trig functions have many similarities and many differences. The graphs of the sine and cosine look very much alike, as do the tangent and cotangent, and then the secant and cosecant. But those three groupings look different from one another.
The one characteristic that ties all these functions together is the fact that they're periodic, meaning they repeat the same curve or pattern over and over again, in either direction along the x-axis. Check out the below figures to see for yourself.
Question: True or False: The graphs of sine and cosine are identical. Answer: False | 677.169 | 1 |
figured that it would be possible to draw a star by figuring out where all the
diagonals of a polygon intersect.
I saw two problems here. First, finding the intersection point of two lines is
a lot of calculation. Not particularly hard calculation, but a lot of it,
and it gets tricky when you have vertical lines.
Second, I couldn't have a four-sided or three-sided
arrow shape; there weren't enough diagonals.
Then, I had another idea. I can't tell you where the idea came from
or how I arrived at it; it just hit me. That's a part of the process
that I don't think can be taught. Here was the idea:
What if you had a
polygon cut out of cookie dough and you sort of pushed in the sides
to make a star shape? That is a method that works for squares and triangles.
When you push in the sides, you push them in at the midpoint so that
you get a nice symmetric cookie. From there, it wasn't a big
leap to figure out: "what if I had a shorter radius at half
of every slice of the polygon?"
This code would be fairly easy to write. I would need one extra
parameter: the proportion of the small radius to the
big radius. In the following code, an if statement
controls whether to use the short radius or the long radius.
I also put in an overloaded version that draws a star with
equal width and height and a start angle of zero. For the
test, I made the short radius one half the long radius.
What Went Wrong?
When I ran this program, I just freaked out. Everything looked great,
except for the three-sided star. How come I didn't
get a star from it? The code sure looks correct, so I decided to see
what would happen if I drew the diagram by hand. I measured
the angles with my protractor,
and I drew the long radius lines with a length of two inches in black
and the short radius lines with a length of one inch in red. Sure
enough, it just so happens that the endpoints of the short radius lines
are right on the lines of the main triangle. The program is drawing a
star, with the sides pushed in by zero.
While wondering why this happened, I remembered that the
cosine of the angle between the lines, 60° (π/3) comes out to 0.5,
and I strongly suspected that this had something to do with it. To
test my hypothesis, I changed the square to use a proportion of
cosine of 45° (π/4), the pentagon to cosine of 36°
(π/5), and the
hexagon to cosine of 30° (π/6). Sure enough, they all came out
with no push-in.
So, if you are drawing a star
with n sides and you set the proportion for the
short radius to long radius to the cos(π/n), you get
a non-star star! I still can't write a mathematical
Question: What was one of the problems the author faced with this initial idea? Answer: Calculating the intersection points of two lines, especially when dealing with vertical lines, was found to be too complex. | 677.169 | 1 |
Cut The Knot!
An interactive column using Java applets
by Alex Bogomolny
The Parabola
March 2004
Menaechmus (c. 375-325 BC), a pupil of Eudoxus, tutor to Alexander the Great, and a friend of Plato [Smith, p. 92], is credited with the discovery of the conics. A more revealing term is conic sections, on account of their being found as the intersections of circular cones by planes. If the planes pass through the vertex of the cone, the conics are said to be degenerate, otherwise they are not. There are three non-degenerate conics: the ellipse, the parabola, and the hyperbola.
The parabola results when the plane is parallel to a generating line of the cone. This is an unbounded curve some of whose properties will be discussed and illustrated below. The plane can be thought of as being hinged on a straight line perpendicular to the axis of the cone. Rotate the plane even a little bit in one direction so that it still cuts only one nappe of the cone, and the intersection will become a closed curve — an ellipse; turn it in the other direction, and the plane will cut the second nappe, so that the curve will acquire a second infinite branch. This 2-piece curve is known as a hyperbola.
Following Pappus (c. 290-350 AD), the common belief is that it was Apollonius of Perga (c, 262-190 BC) who gave the curves their names.
The term parabola comes [Schwartzman, p. 158] from Greek para "alongside, nearby, right up to," and -bola, from the verb ballein "to cast, to throw." Understandably, parallel and many of its derivatives start with the same root. The word parabola may thus mean "thrown parallel" in accordance with the definition.
The story is interesting, and a short etymological regression won't be entirely out of place. As I. Kant once appropriately (see below) remarked, "... even clever persons occasionally chatter" [Kant, p. 55].
Question: What does the term 'parabola' mean etymologically? Answer: "Thrown parallel"
Question: Who is credited with the discovery of the conics? Answer: Menaechmus
Question: What happens when the plane is rotated in one direction so that it still cuts only one nappe of the cone? Answer: The intersection becomes a closed curve - an ellipse | 677.169 | 1 |
The applets below illustrate several purely geometrical properties of the parabola. For entirely idiosyncratic reasons, the parabola has been rotated 90o such that wherever a parabola had to be drawn, I used the equation y = x2/2p instead of (2). In the following, the feet of perpendiculars dropped from points A, B, etc. on the parabola to the directrix will be denoted A', B', etc. F will always denote the focus of the parabola at hand.
Theorem 1
Let A lie on a parabola. Then the tangent to the parabola at A makes equal angles with AF and AA'.
Proof
By the definition, FAA' is isosceles. Let T be the midpoint of FA'. Then the perpendicular bisector AT divides the plane into two parts: one consists of points that are nearer to F than they are to A'; the other consists of points that are nearer to A'. Except for A, all points of the parabola lie in the former half. Indeed, let B be a point on the parabola. Then, since BB' is the shortest segment from B to the directrix, FB = BB' < BA'. In particular, B does not belong to AT. We conclude that A is the only point of intersection of that line with the parabola. Therefore, AT is tangent to the parabola at A.
Corollary (Parabolic mirror)
If a light source is placed at the focus of parabola and the light is reflected from its inner surface, the reflected rays are all parallel to the axis. Radio telescopes are built on a reversed principle. Incoming signals parallel to the axes all pass through the focus.
If, at the outset, the parabola is not given, but only a point and a line, we may produce any number of creases by folding the paper so that the given point falls onto the given line. In time, a parabola will emerge as the envelope of paper lines. Note that it does not matter whether the goal of a particular folding is to place a point on a line, or make the line pass through the point. As a practical matter, if the given line coincides with a paper edge then it is much easier to pursue the latter goal.
The parabola divides the area of an Archimedes triangle in the ratio 2:1. In other words, the area of the parabolic segment AB equals 2/3 of the area of the Archimedes triangle ABS.
Proof
Let area(ABS) = 1. Two thousand years before the invention of Calculus, Archimedes filled the parabolic segment with triangles, whose areas are easily arranged into a geometric series whose sum he already knew.
The first triangle in the series is the "inner triangle" ABO. From Lemma, area(ABO) = 1/2 — the first term of the series. Area(A1B1S) = 1/4. Therefore, area(AA1O) + area(BB1O) = 1/4.
Question: What is the ratio in which the parabola divides the area of an Archimedes triangle? Answer: 2:1 | 677.169 | 1 |
Proposition 34
To find two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.
Set out two medial straight lines AB and BC, commensurable in square only, such that the rectangle which they contain is rational and the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB.
Question: What kind of rectangle do lines AB and BC contain? Answer: The rectangle contained by lines AB and BC is rational. | 677.169 | 1 |
The three volcanic peaks
on Easter Island form an isosceles triangle with base
angles of 36°. The relationship between the length of the base of
this triangle and the lengths of the sides precisely expresses
phi:
1.95 x 1.618 = 3.16
The Great Pyramid is
10,055 miles from the Southwestern volcanic peak on Easter Island. A
triangle formed by the great circle distances between Easter Island,
the Great Pyramid, and the axis point for the line of ancient sites
in S. E. Alaska, has the same angular dimensions as the triangle
formed by the volcanic peaks on Easter Island, and also precisely
expresses phi:
6, 215 x 1.618 = 10,055
The large triangles in
five pointed stars have these same angular dimensions, and express
the same phi relationship.
All of the angles in five
pointed stars, 36°, 72° and 108°, are numbers that have been
associated with the precession of the equinoxes, as
well as the internal designs of the Great Pyramid and
other ancient sites around the world.
There are 72 major monuments at Angkor, 108
stone figures surround
Angkor Thom, and the highest terraces of Angkor Thom
house 72 Buddhas in latticework stupas.
The great circle distances between Angkor Thom, the ziggurat at Ur, and the axis point of the line of ancient
sites, forms an isosceles triangle with base angles of 72°.
The small triangles in five pointed stars have these same angular
dimensions, also precisely expressing
phi:
3,841 x 1.618 = 6,215.
These relationships are also clearly demonstrated by the metric
system, which is based on a distance of 10,000 kilometers from the
North Pole
to the Equator. The distance from Easter Island
to the
Great Pyramid
is 16,180 kilometers and the distance from Ur to
Angkor is 6,180 kilometers:
After the discovery of America by Europeans, there were early reports
an abandoned city of megalithic stone construction in the Amazon
rainforest.
The legendary golden city of El Dorado was
also believed to be hidden under the rainforest. Explorers such as
Percy Fawcett and
Sir Walter Raleigh have searched for these sites, but their location(s) remain unknown.
Along the line of ancient
sites, the great circle distance from the
Great Pyramid to 4° 10' South Latitude, 56° 19'
West Longitude, is 6,215 miles (exactly 25% of the circumference of
the Earth).
The distance from these
sites to the axis point in Southeast Alaska is also 6,215 miles,
forming a perfect equilateral triangle.
Question: Who were some of the explorers that searched for the legendary golden city of El Dorado? Answer: Percy Fawcett and Sir Walter Raleigh | 677.169 | 1 |
Posts Tagged 'geometry class online':
Nowadays, we would find our life easier because so many online courses like geometry online course that could teach you geometry without required us to physically attend geometry class at school. Actually, What we need is just a desktop with the fast internet connection then we could always jump to learn the online geometry course. It is not like few years back that we have to physically attend geometry class (imagine the time that we have to spend for attending the class and comeback home).
Now with this geometry online class, we have a chance to learn geometry lessons without leaving our home.
Below is the statement that I try to solve one of the geometry problem, even this is not one of the real online geometry courses, but my expectation is to help others to understand the blue print of the simple online geometry course.
Before we jump to the complex of the geometry problem, I would like to play one piano song for you and let your mind relax and it would be easier for you to absorb the geometry lessons as below :
Let's Start :
There is a triangle with <A, <B, <C. Assume the "<" is angle. ¼<C more 12o than (2x<A)-2o. ½<A less 5o than (1:<B)x2o. 2<B more 6o than <C-10o. Then I ask you how much <A ? (The answer may use the fraction)
The answer:
First Equation:
¼<C=2<A-2o
Instant result:
<C=8<A-8o
Second Equation:
½<A = (1:<B)x2o-5o
Instant result:
<A=<B-2½o
Third Equation:
2<B=<C-10o+6o
Instant result:
<B=½<C-2o
Make substitution <C to make it <A!
Because<C=8<A-8o, so the result is:
½<C-2o=4<A-2o(<B=½<C-2o=4<A-2o)
Now, we have three equation, just combine it! We already knew the triangle formula is the three angle, if we combine, the result will always 180o, same like our <A,<B,<C, the formula is :
<A+<B+<C=180o
Now, we come to counting again…
<A=<A , <B=4<A-2o , <C=8<A-8o
Combine them with the statement the result must 180o
<A+4<A-2o+8<A-8o = 180o
13<A-10o = 180o
<A = 14 8/13o
Question: Is it possible to learn geometry without physically attending a geometry class at school? Answer: Yes, it is possible due to the availability of online geometry courses. | 677.169 | 1 |
Understanding Angles
Polygon calculations come up frequently in woodworking. Finding the angles and dimensions of used in building multi-sided frames, barrels and drums (to name a few applications) begins with an understanding to the geometry of regular (symmetrical) polygons.
Figure 1
Regular Polygon Shapes: Calculating the Bevel or Miter Angle of the Parts In most cases, to build a polygon shape, you'll need to know the bevel or miter angle necessary to join the sides. To do that, you'll need to use trigonometric functions in conjunction with a basic property of the polygon shape.
A regular polygon is an example of a complex shape that can be thought of as the splicing together of a number of right triangles. Specifically, a regular polygon with N sides can be divided into N * 2 "fundamental" right triangles. The 6-sided polygon in Figure 1, for example, can be divided into 12 equally proportioned right triangles.
The lines that form two sides of each triangle also cut through the center of the circle that circumscribes the polygon. Because the lines divide the circle into equal sections, we know that each triangle will have one acute angle equal to 360/(N * 2).
Figure 2
For angle a of triangle AOH in Figure 2:
a = 360/(6 * 2)
=>
a = 30 degrees
Angle b of triangle AOH is the complement of angle a:
b = 90 - 30
=>
b = 60 degrees
The saw setting necessary to cut the bevel or miter where the joints meet will be either angle a or angle b, depending on the calibration system of the saw you are using. Cutting the beveled edge of barrel or drum staves on a table saw will almost always require you to set the saw at angle a, because nearly all table saw bevel angle scales are calibrated to treat a straight up and down vertical setting of the blade as a 0 degree setting. The same would hold true if you are cutting the parts for a multi-sided frame on a miter saw. Most table saw miter gauges, on the other hand, treat a square cut as a 90 degree setting, making angle b the correct angle setting.
Calculating the Dimensions of a Polygon So far, we know the how to calculate the acute angles that make up the fundamental right triangles of a polygon with N sides. Now, using that information, we can find the dimensions of the parts we'd need to cut to build a polygon shape, based on the overall dimensions of the polygon or, if we wanted to, calculate the overall dimensions of the shape based on the dimension of the sides of the polygon.
Question: What is the relationship between the angles 'a' and 'b' in the fundamental right triangles of a regular polygon? Answer: Angle b is the complement of angle a. | 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: Which two words does the user use to describe their request for an explanation? Answer: Adequate, response | 677.169 | 1 |
c = b = g = f
a = d, e = h, c = b, and g = f because they are vertical angles.
d = e and c = f because they are opposite interior angles.
a = e, d = h, c = g and b = f because they are corresponding angles.
Now you're ready to tackle any geometry questions asking about parallel angles and lines.
Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.
For more SAT advice from Veritas Prep, watch "SAT Tip: How The Calculator Can Hurt Your Score On the SAT Math Section"
Created for Bloomberg Businessweek readers, this diagnostic quiz is designed to measure your ability level with 25 realistic SAT questions. Click here to take the quiz and get instant feedback about your performance.
Created for Bloomberg Businessweek readers, this diagnostic quiz is designed to measure your ability level with 25 realistic GMAT questions. Click here to take the quiz and get instant feedback about your performance.
Question: Where did Vivian Kerr study abroad? Answer: London. | 677.169 | 1 |
Midpoint of A and B is X=(A+B)/2. The vector to that point is OX, which is also (OA+OB)/2
The diagonals in a parallelogram with sides......
sum and diff of 2D arrays ..., a should contain the sum of the original arrays (elementby- element), while b should contain the difference. The function should not return any value.*/
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answers with short explanations ...nd a vice chairperson be selected from a committee of 12 senators? 12*11 ways
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In a recent survey of 100 women, the following information was gathered.
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Detailed Answers to all Questions ...and x-y=7
5.3.8. A chemist who has five assistants is engaged in a research project that calls for nine compounds that must be synthesized. In how many ways can the chemist assign these......
Answers ...ion 2
Suppose A is the set of students currently registered at the University of Calgary, B is the set of professors at theUniversity ofCalgary, and C is the set of......
Detailed Answer - Alternating Series ...n
The series 'ąĎ i=1 ¬†'ąě (-1) (i+1) ¬†(i+3)/(i 2+10)¬† is an alternating series because (i+3)/(i 2+10) 'Č•0 for all i 'Č•1.
Consider the difference between two consecutive elements in the series:
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Conics ...his is what we should do.
1. First rewrite the equation in the form Ax^2+Bxy+Cy^2+Dx+Ey+F=0
The coefficients A,B,C,D,E,F can be 0, but at least one of A,B,C would be non-zero.
2. Now calculate the......
Counter Value step-by-step ...†¬†¬†¬†¬† // So, the first time the inner loop goes from 3 to 5, then from 3 to 6 and so on, until it goes from 3 to s
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c) If G is a self-complementary graph on n vertices, where n>1, prove that n=4k or n=4k+1, for some K that is a positive integer.
Please include the detailed explanation for every step of the......
Question: In a survey of 100 women, how many use either Shampoo A or Shampoo B or both?
Answer: To find this, we use the principle of inclusion-exclusion: 47 (Shampoo A) + 37 (Shampoo B) - 21 (both) = 63 women use either Shampoo A or Shampoo B or both.
Question: How many ways can a chemist assign 9 compounds to 5 assistants?
Answer: This is a combination problem, as the order of assignment does not matter. The number of ways is C(9,5) = 126. | 677.169 | 1 |
Directions: Use your computer and projector to launch the Vec-->Touring web site. Use the projector to show the students how to use the controls using an easy example. Point out the controls on the left including the volume adjustment (very handy). The controls for entering vectors are a little strange. For example, to move 3 units to the right and 2 units down the student would have to:
click the "X" button, the "right arrow", and the "3"
followed by
click the "Y" button, the "down arrow", and the "2".
Then hit the "Go" button.
Have the students do one easy example before they try the more difficult examples. The more difficult ones have decimals. Most students will begin the easy examples by counting the number of boxes horizontally and vertically. When they get some practice they will use the location controls in the lower left and then add or subtract accordingly.
After about 30 minutes of playing have the students quit the simulation and discuss how slope is defined as rise over run. Ask the students how they found rise and run on the simulation. Emphasize that left arrow and down arrow are negatives when graphing or finding slope. This is also a good opportunity to introduce the slope formula since the students were using most of it during the simulation.
Question: What is the final button to press after entering the vector coordinates? Answer: The "Go" button | 677.169 | 1 |
Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.
Thébault's problem II
Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.
Thébault's problem III
Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.
This Saturday, November 8th, Simion Filip will present the way our understanding of the physical world shaped the geometric problems that we considered throughout history. Examples will be drawn mostly from elementary Euclidian geometry and the talk should be accessible to anyone who is familiar with angles and triangles.
Mr. Filip is a senior at Princeton University, studying mathematics with an interest in mathematical physics. After graduation, he plans to pursue a Ph. D. in mathematics. While in high-school, Simion Filip took part in both the International Mathematics and Informatics Olympiads, where he received silver and bronze medals respectively.
Question: What is the relationship between the quadrilateral formed by the centers of the four external squares on the sides of a parallelogram? Answer: It is a square. | 677.169 | 1 |
What is a Composite Figure?
Note:
Ever notice that some figures look like a combination of multiple other figures? These types of figures are called composite figures. This tutorial introduces you to composite figures and shows you how to break up a composite figure into multiple shapes. Take a look!
Did you know that the formula for the area of a triangle can be found by using the formula for the area of a parallelogram? In this tutorial, you'll see how it's done!
Question: What is the relationship between the area formulas of a triangle and a parallelogram, according to this text? Answer: The text states that the formula for the area of a triangle can be found using the formula for the area of a parallelogram. | 677.169 | 1 |
Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 bce). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th century, when non-Euclidean geometries attracted the attention of mathematicians, geometry meant Euclidean geometry. It is the most typical expression of general mathematical thinking. Rather than the memorization of simple algorithms to solve equations by rote, it demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts, and an insistence on the importance of proof. In Euclid's great work, the Elements, the only tools employed for geometrical constructions were the ruler and the compass—a restriction retained in elementary Euclidean geometry to this day.
In its rigorous deductive organization, the Elements remained the very model of scientific exposition until the end of the 19th century, when the German mathematician David Hilbert wrote his famous Foundations of Geometry (1899). The modern version of Euclidean geometry is the theory of Euclidean (coordinate) spaces of multiple dimensions, where distance is measured by a suitable generalization of the Pythagorean theorem. Seeanalytic geometry and algebraic geometry.
Fundamentals
Euclid realized that a rigorous development of geometry must start with the foundations. Hence, he began the Elements with some undefined terms, such as "a point is that which has no part" and "a line is a length without breadth." Proceeding from these terms, he defined further ideas such as angles, circles, triangles, and various other polygons and figures. For example, an angle was defined as the inclination of two straight lines, and a circle was a plane figure consisting of all points that have a fixed distance (radius) from a given centre.
As a basis for further logical deductions, Euclid proposed five common notions, such as "things equal to the same thing are equal," and five unprovable but intuitive principles known variously as postulates or axioms. Stated in modern terms, the axioms are as follows:
1. Given two points, there is a straight line that joins them.
2. A straight line segment can be prolonged indefinitely.
3. A circle can be constructed when a point for its centre and a distance for its radius are given.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.
1. For any two different points, (a) there exists a line containing these two points, and (b) this line is unique.
5. For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique.
Question: What is the main focus of Euclidean geometry, unlike other mathematical subjects? Answer: It demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts, and an insistence on the importance of proof | 677.169 | 1 |
The fifth axiom became known as the "parallel postulate," since it provided a basis for the uniqueness of parallel lines. (It also attracted great interest because it seemed less intuitive or self-evident than the others. In the 19th century, Carl Friedrich Gauss, János Bolyai, and Nikolay Lobachevsky all began to experiment with this postulate, eventually arriving at new, non-Euclidean, geometries.) All five axioms provided the basis for numerous provable statements, or theorems, on which Euclid built his geometry. The rest of this article briefly explains the most important theorems of Euclidean plane and solid geometry.
Plane geometry
Congruence of triangles
Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and the congruence theorems specify the conditions under which this can occur. The first theorem illustrated in the diagram is the side-angle-side (SAS) theorem: If two sides and theThe first very useful theorem derived from the axioms is the basic symmetry property of isosceles triangles—i.e., that two sides of a triangle are equal if and only if the angles opposite them are equal. Euclid's proof of this theorem was once called Pons Asinorum ("Bridge of Asses"), supposedly because mediocre students could not proceed across it to the farther reaches of geometry. (For an illustrated exposition of the proof, seeSidebar: The Bridge of Asses.) The Bridge of Asses opens the way to various theorems on the congruence of triangles.
The parallel postulate is fundamental for the proof of the theorem that the sum of the angles of a triangle is always 180 degrees. A simple proof of this theorem, attributed to the Pythagoreans, is shown in the diagram.
As indicated above, congruent figures have the same shape and size. Similar figures, on the other hand, have the same shape but may differ in size. Shape is intimately related to the notion of proportion, as ancient Egyptian artisans observed long ago. Segments of lengths a, b, c, and d are said to be proportional if a:b = c:d (read, a is to b as c is to d; in older notation a:b::c:d). The fundamental theorem of similarity states that a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle's third side.
The similarity theorem may be reformulated as the AAA (angle-angle-angle) similarity theorem: two triangles have their corresponding angles equal if and only if their corresponding sides are proportional. Two similar triangles are related by a scaling (or similarity) factor s: if the first triangle has sides a, b, and c, then the second one will have sides sa, sb, and sc. In addition to the ubiquitous use of scaling factors on construction plans and geographic maps, similarity is fundamental to trigonometry.
Question: Which of the following is NOT an axiom in Euclid's geometry? A) The sum of angles in a triangle is 180 degrees B) Two points determine a line C) Through a point not on a line, there is exactly one line that does not intersect the given line D) All right angles are equal Answer: A) The sum of angles in a triangle is 180 degrees
Question: Which theorem is often referred to as the "Bridge of Asses"? Answer: The basic symmetry property of isosceles triangles
Question: What is the name given to the theorem that states two triangles are congruent if their corresponding sides are proportional? Answer: The AAA (angle-angle-angle) similarity theorem | 677.169 | 1 |
The angles being: obtuse (bigger than 90 degrees), accute (smaller than 90 degrees), and right (90 degrees exactly). The triangles being equilateral (all 3 sides are equal in length), isosceles (2 sides are equal in length), and scalene (no sides equal in length). We also covered right triangles…the special kind of polygon where two of the sides form a right angle.
Was that as boring to read as it was to type?! Yes, methinks so. So, I'll tell you how we talked about angles and triangles the fun way…using our bodies.
right angle, using your arms:
Acute angle, using your arms, arm, or fingers:
Once the kids got comfortable making angles, they started making them with whatever part of their bodies they could. Note that there are three different models of acute angles: Naturalist is using one arm and bending it way closed to make her angle…her friend T is using two fingers spread only a little wide to make his, while golfer is doing the two armed style. Later on they used legs, feet, wrists..anything with a hinge.
After a few warm ups to get everyone on board the angle train, we started Simon Says…Simon Says make an acute angle, Simon Says make an obtuse angle, Simon Says make a right angle with your leg, Simon Says make an acute angle with your arm…etc. etc.:
As you can see, this is multi-age friendly. Lots of laughs were had.
With only 3 angles, it didn't take long for that part of it to be over with, so then we moved on to the triangle part of it. All the kids were in one group, and when I named a triangle they had to choose who would help make it with their body. For instance, I said, "Equilateral Triangle" and even though I'm pretty sure none of my kids could give the definition correctly, they knew that they had to find 3 kids that were the same height. It's just another way of processing information…they don't really connect to a triangle on a sheet of paper, and having 3 equal sides doesn't really matter to them…until they have to construct one using their friends.
So here's as close as we could get to equilateral:
They tried to build an isosceles triangle standing up, which didn't really work…it looked more like a pentagon:
so they had to reform to make it on the ground:
I have to say, one of the most fun parts was rolling the younger kids into position…Sassy in particular is very ticklish:
Most interesting of all was making a right triangle. We knew, from the Pythagorean experiments we'd done the last week, that important numbers were 3,4, and 5…because if you take the squares of 3, 4, and 5 you end up with a right triangle. The kids set about finding a similar ratio that applied as a relationship to their sizes, and decided that my three would be perfect to make it:
Question: How many angles were covered in the activity? Answer: Three (acute, obtuse, and right).
Question: What is the definition of an acute angle? Answer: An angle smaller than 90 degrees. | 677.169 | 1 |
The Modern Day High School Geometry Course: A Lesson in Illogic
by Barry Garelick Chances are good that most students in high school today know that the sum of the measures of angles in a triangle equals 180 degrees. Unfortunately, chances are also good that most high school students today cannot prove that proposition. Geometry as taught today is for the most part lacking in the [...]
by Barry Garelick
Chances are good that most students in high school today know that the sum of the measures of angles in a triangle equals 180 degrees. Unfortunately, chances are also good that most high school students today cannot prove that proposition.
Geometry as taught today is for the most part lacking in the most important aspect of the subject: Proofs. Prior to 1980, most if not all high school geometry classes were very much proof-based. While there are those who bemoan the teaching of K-12 math as being mired in "computational" and "procedural" aspects of math while ignoring the larger beauty of what mathematics is about, it is ironic that when it comes to geometry, the true mathematical nature of the subject is largely ignored.
A glance at the geometry textbooks that are typically used in high schools today reveals that the problems students are given in such courses require one or two proofs that are not very challenging in a set of problems devoted to the application of theorems rather than the proving of propositions. Most of the problems presented in these textbooks require students to apply various theorems and definitions to find the lengths of line segments and angles. Typical courses in geometry are lacking in proof-based problems; instead, they contain many problems in which missing angles or segments are indicated as algebraic expressions. For example, opposite sides of a quadrilateral that is identified as a parallelogram may be labeled x + 2 and 2x – 6; the student is asked to find the length of the segments. This problem requires knowledge of the properties of a parallelogram leading to the conclusion that the two segments of interest are congruent. The two sides, expressed as x + 2 and 2x – 6 then lead to the equation x + 2 = 2x – 6. Figure 1 shows another example of a problem that does not require formal proof.
Figure 1:
This problem requires students to know and apply that the sum of the angles in a triangle equals 180 degrees, and to know what are linear pairs of angles, and that they sum to 180 degrees. From this, students can piece together information and compute angle R.
While the types of problem discussed above constitute a form of proof (requiring applications of theorems and definitions combined with deductive reasoning to justify the necessary computation), such problems do not fully develop the skills necessary to develop a logical series of statements that constitute proof. In contrast, consider a problem that requires a student prove a particular proposition, such as shown in Figure 2:
Figure 2:
This problem does not require any numerical calculation. It requires knowledge of theorems of parallel lines in a plane and properties of isosceles triangles.
Defeating the Purpose of a Geometry Course
Question: What type of problems are typically found in high school geometry textbooks today? Answer: Problems requiring application of theorems and definitions to find lengths of line segments and angles
Question: What is the difference between the problems in Figure 1 and Figure 2? Answer: Figure 1 requires computation, while Figure 2 requires a logical series of statements to prove a proposition
Question: What is the purpose of a geometry course, according to the text? Answer: To develop the skills necessary to develop a logical series of statements that constitute proof | 677.169 | 1 |
When you insert a rectangle or a callout box using the drawing tools and activate the function Edit Points, you see a small frame at the upper left corner of the object.The frame indicates the amount by which the corners are rounded
AndShowing page 1. Found 143 sentences matching phrase "-cornered polygon".Found in 5.4 ms. Translation memories are created by human, but computer aligned, which might cause mistakes. They come from many sources and are not checked. Be warned.
Question: What is the relationship between the frame and the corners of the object? Answer: The frame indicates the amount by which the corners are rounded | 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 a double ordinate of a parabola? Answer: Any chord of the parabola which is perpendicular to the axis is called a double ordinate.
Question: What is the tangent at the vertex of a parabola? Answer: The straight line perpendicular to the axis of the parabola passing through the vertex is called the tangent at the vertex. | 677.169 | 1 |
With this virtual manipulative, students arrange sides and angles to construct congruent triangles. They drag line segments and angles to form triangles and flip the triangles as needed to show congruence. Options include constructing triangles given three sides (SSS), two sides and the included angle (SAS), and two angles and an included side (ASA). But the option that will motivate most discussion is constructing two triangles given two sides and a nonincluded angle (SSA). The question in this case is: Can you find two triangles that are not congruent?
Transformations—Reflections
Here students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle.
The Pythagorean Theorem
This site invites learners to discover for themselves "an important relationship between the three sides of a right triangle." The site's author, Jacobo Bulaevsky, speaks directly to students, encouraging them throughout five interactive exercises to delve deeper into the mystery. Within each exercise he gives hints that will motivate and entice your students 12/09/2011.
Question: What can students do with the transformations tool? Answer: Manipulate geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. | 677.169 | 1 |
An angle measured from a vertical reference. Zero degrees is a vertical line pointing up, 90 degrees is horizontal, and 180 degrees is straight down. Surveyors' Slang Surveying, like any profession, has its special terms and slang. Some are just humorous, some help distinguish similar sounds (e.g. eleven and seven), and some are just plain strange
This indicates how the airfoil is inclined to the air coming towards it. It is measured in degrees. At zero degrees, the front of the airfoil is pointed directly into the stream. At 90 degrees it is pointed straight up. It is also referred to as the Angle of Attack. The angle of attack in this simulation is limited since it does not depict stall characteristics.
(plane angle) The spatial relationship between two straight lines which meet; the point of meeting is the vertex of the angle. Angles are measured in degrees or, alternatively, in radians. A complete revolution is 360 degrees (360°). If two lines are parallel, the angle between them is zero. A straight line forms an angle of 180° and a right angle is 90°.
a configuration of two line segments meeting at a point. The term is often used for the measure of rotation from one of the line segments to the other. In this sense, a right angle measures 90°, an acute angle is less than 90°, an obtuse angle is greater than 90° but less than 180°, and a reflex angle is greater than 180°.
(1) A figure consisting of two rays with the same endpoint. The endpoint is called the vertex of the angle. An acute angle has a measure greater than 0° and less than 90°. An obtuse angle has a measure-ment greater than 90° and less than 180°. A right angle has a measurement of 90°. A straight angle has a measure of 180°.
The angle of the dot is the angle at which the dots chain together. The problem with most computer graphics programs is that the angles of the halftones are generally great for offset printing but not good for screening. A lot of computer programs use 45 degrees as the default angle. Actually, 20 to 25 degrees is good for basic halftone work. If you are doing a process color job you can try Cyan 15, Magenta 45, Yellow and Black 75, or Cyan 22.5, Magenta 52.5, Yellow and Black 82.5.
Two rays with a common endpoint. The common endpoint is called the vertex of the angle. An acute angle has a measure greater the 0° and less than 90°. An obtuse angle has a measure greater than 90° and less that 180°. right angle has a measure 90°. straight angle has measure 180°. See also reflex angle.
Question: What is the angle between two parallel lines? Answer: Zero degrees | 677.169 | 1 |
construct a median
Answers
Construction:
1. With one endpoint of the line as a center and a radius with more than 1/2 the length of the line, construct arcs above and below the line.
2. With the other endpoint of the line as a center using the same radius, construct similar arcs that intersect the previously constructed arcs.
3. Draw a line segment connecting the intersection points of the constructed arcs.
Question: What is the first step in constructing a median? Answer: With one endpoint of the line as a center and a radius more than half the length of the line, construct arcs above and below the line. | 677.169 | 1 |
Trigonometry-basics/684424: Solve the triangle using the given information.
A=105degrees
b=12
c=9
I'm pretty sure you need to use law of sines: because your given two sides and one angle.
But I don't know how to start. Please provide full steps 1 solutions Answer 424085 by jim_thompson5910(28504) on 2012-11-24 18:56:10 (Show Source):
Equations/684156: Show that the points A (3,-5), B (-3,-1) and C (7,1) are vertices of an isosceles triangle. Find the area? Help pls. with graph and solution! 1 solutions Answer 423939 by jim_thompson5910(28504) on 2012-11-23 17:14:35 (Show Source):
Graphs/684114: Describe in words how you would obtain the graph of y = 3(x-4)^2 from transmission of the graph y = x^2 make sketches of both parabolas on the same set of axis.
can you please explain this to me? thanks in advance 1 solutions Answer 423911 by jim_thompson5910(28504) on 2012-11-23 14:41:46 (Show Source):
So the solution is x=20. Don't forget to check it (I'll let you do this).
Quadratic_Equations/683994: A programmer is writing the code for a new interactive basketball game. As a result, she is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modeled by the quadratic relation,h= -0.2d^2 + 3d + 6 , where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.
How high was the ball when the shooter shot it?
What was the maximum height obtained by the ball?
A rim of a basketball net is 10 feet high. For what horizontal distance was the ball above the rim of the basketball net?
Please help for my review coming up ): 1 solutions Answer 423860 by jim_thompson5910(28504) on 2012-11-22 20:49:04 (Show Source):
Question: Which points are the vertices of the isosceles triangle? Answer: A (3,-5), B (-3,-1), and C (7,1)
Question: For what horizontal distance was the ball above the rim of the basketball net, given that the rim is 10 feet high? Answer: 16 feet | 677.169 | 1 |
Examples quadrants's examples
Learn about Quadrants on . Find info and videos including: How to Use a Navigational Quadrant, How to Convert Quadrants in Maths, How Do Quadrants Work in Math? and much more. — "Quadrants - ",
Posted by Webminister in District News, Quadrants | No Comments Monday Morning Quadrant Meetings are cancelled on March 15 for Spring Break and April 5 for Easter Holiday. However everyone is. — "Quadrants " North District UMC",
The x- and y- axes divide the coordinate plane into four regions. These regions are called the quadrants. — "Quadrants - Math Dictionary",
Quadrants' definition, a quarter of a circle; an arc of 90°. See more. — "Quadrants' | Define Quadrants' at ",
Definition of Quadrants in the Online Dictionary. Meaning of Quadrants. Pronunciation of Quadrants. Translations of Quadrants. Quadrants synonyms, Quadrants antonyms. Information about Quadrants in the free online English dictionary and. — "Quadrants - definition of Quadrants by the Free Online",
Subject > Math > Get Geometry Help, Online Geometry Tutoring > Cartesion System > Quadrants The table below shows the sign of the coordinates of the point in different quadrants. The following table shows the position of some points in the quadrants. Example:. — "Cartesian System, Four Quadrants | ",
Quadrant -- from Wolfram MathWorld. Quadrant. One of the four regions of the plane defined by the four quadrant system which is in theory also known as the system of centre. — "Mathematical Quadrants",
Building throttle quadrants over 45 years. 128 models available in four sizes: 1/2, 3/4, standard and large. Single arm to three arm throttle quadrants with friction control. Twin engine quadrants Single function to three control six arm. — "Aircraft Throttle Quadrant by Baxter Throttle Quadrants",
FCS Express supports three different kinds of quadrants: standard quads, non-rectangular quads called floating quadrants, and percentile quadrants. Standard quadrants are the traditional type of quadrants: a crosshairs which divides the plot into 4 rectangular sections. — "De Novo Software",
The implication is that in holistic mathematical terms the interpretation of quadrants involves the combination of both linear and circular notions of understanding. Secondly the directions of the lines vary so that those in opposite quadrants have different signs. — "The Four Quadrants", iol.ie
Covers the terminology and notation for the four quadrants of the plane, and answers some typical homework problems related to quadrants. — "The Quadrants of the Cartesian Plane",
Question: Which of the following dates are the North District UMC's Monday Morning Quadrant Meetings cancelled? A) March 15 B) April 5 C) Both D) Neither Answer: C) Both
Question: How many models of aircraft throttle quadrants does Baxter Throttle Quadrants offer? Answer: 128 | 677.169 | 1 |
Trapezoids: Finding Angles and Segments.
Do you know that when you have an isosceles trapezoid: All acute angles are congruent, all obtuse angles are congruent, and that if you take one acute, and one obtuse at a time they are supplementary?
In this lesson, you will work with isosceles trapezoids. You will view the solution of the problems with an emphasis in highlighting the concepts from previous sections that apply to this type of problems. You will be given a few problems embedded in the lessons to test your understanding. Venture yourself to a new trapezoid experience!
Angle
Geometry shape formed by two rays (initial and ending sides of the angle) that share a common endpoint called the vertex. You may name an angle using the vertex, or a point in each ray and the vertex label in the center.
Isosceles trapezoid
Trapezoid with two non-congruent and non-parallel sides.
Polygon
It is a closed plane figure with a least three straight segments as sides.
Quadrilateral
A four-sided polygon.
Segment
Line segment; A section of a line, defined by two end points and all the points between them.
Trapezoid
Quadrilateral with exactly one pair of parallel sides.
Interactive Geometric Applets: Relevant Theorems.
In an Isosceles Trapezoid opposite angles are supplementary, it has only one pair of parallel sides,
and one pair of congruent sides. Diagonals are congruent, but they don't bisect each other.
This interactive geometric applet will allow you to visualize these properties in a dynamic
way.
When working with isosceles trapezoids the parallel segments form
consecutive interior angles that are supplementary. You may review this property
Question: What does it mean for a trapezoid to be isosceles? Answer: It means a trapezoid with two non-congruent and non-parallel sides. | 677.169 | 1 |
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