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Now, the very last thing I want to tell you about these functions is - well, the values at certain famous points. A lot of times you want to find the values at some big point - let me tell you the values that you should know for certain. You should know - I'll write them first in degrees. You should the trig functions at zero degrees, at 30 degrees, at 45 degrees, at 60 degrees, and at 90 degrees. You should know all the trig value functions just off the top of your head for those values, at least. Now let me convert them to the language of Calculus. Let me convert those to radians. This would be zero radian, and you could try this on your own by just looking at conversion that pi equals 180. This is going to give you pi over six radian. This will give you pi over four radian, like we saw before. This will give you pi over three radian. This will give you pi over two; these are all radian. And if you don't believe me here for example, take pi over three and now multiply that by 180 over pi. If you imagine putting in a 180 over pi here - let's just do that one for fun, so you can see that. I put in a 180 over pi. Well, you see the pi's cancel and three goes into 180, 60 times. So in fact you see 60 degrees.
Question: Which of the following is NOT a conversion given in the text?
A) 0 degrees to radians
B) 30 degrees to radians
C) 90 degrees to radians
D) 180 degrees to radians
Answer: D) 180 degrees to radians.
Question: What is the conversion of 60 degrees to radians?
Answer: Pi over three radians (π/3). | 677.169 | 1 |
The algorithm manipulates a list of red and blue isosceles triangles. Each red triangle has a 36° angle at its apex, while each blue triangle has a 108° angle.
In Python, we can represent such triangles as tuples of the form (color, A, B, C). For the first element, color, a value of 0 indicates a red triangle, while 1 indicates blue. The rest of the tuple gives the co-ordinates of the A, B and C vertices, expressed as complex numbers. Complex numbers work well here since they can represent any point on the 2D plane — the real component gives the x co-ordinate, while the imaginary component gives the y co-ordinate.
As you can see, we draw an outline along the sides of the triangle, but not along the base. This allows each triangle to connect with another triangle of the same color, forming the rhombus-shaped tiles that are visible in the final Penrose tiling.
Now here's the fun part. Given a list of such triangles, we can subdivide each one to generate another triangle list. A red triangle is subdivided into two smaller triangles as follows:
The above subdivision introduces a new vertex P, located at a point along the edge AB which satisfies the golden ratio, .
Similarly, each blue triangle is subdivided into three smaller triangles:
This subdivision introduces two new vertices: Q along the edge BA, and R along the edge BC, at points which also satisfy the golden ratio. As well, two of the resulting triangles are mirrored — I've drawn a highlight in the corner of each triangle to help identify which ones are mirrored and which are not.
All of the above steps can be performed using just a few lines of Python. This function accepts a list of triangles represented as tuples, subdivides each one, and returns the new triangle list:
Using all of the above code, we can, for example, start with a single red triangle, subdivide it several times, and draw the result after each subdivision. You can see the tiling pattern begin to emerge:
You can even begin the sequence using another triangle list. Here's some code to start with a "wheel" shape consisting of 10 red triangles:
If we subdivide this wheel shape repeatedly, we get the following sequence of tilings. Notice that each tiling contains a lot of symmetry — both reflective and rotational symmetry around 5 different axes:
If you study either the top or bottom row of this sequence carefully, you'll notice that for each tiling except the first, an upside-down copy appears in the tiling to the right. I've drawn some yellow outlines to make this more obvious. Looking at it another way: if you take any of these tilings, subdivide it twice, flip it vertically and enlarge the result, you've basically added another ring around the tiling. By repeating this process indefinitely, you can see how a Penrose tiling could be made to completely fill the entire plane.
Question: What is the total number of triangles in the initial wheel shape used in the example? Answer: 10
Question: Which of the following is NOT a step in subdividing a red triangle? A) Introducing a new vertex P B) Mirroring two of the resulting triangles C) Introducing two new vertices Q and R Answer: C) Introducing two new vertices Q and R
Question: How many new vertices are introduced when a blue triangle is subdivided? Answer: 2
Question: What is the angle at the apex of a red triangle? Answer: 36° | 677.169 | 1 |
Given a set of points, the plane can be split in domains for which the first point is closest, the second point is closest, etc. Such a partition is called a Voronoi diagram. If one draws a line between any two points whose Voronoi domains touch, a set of triangles is obtained, known as the Delaunay triangulation. Generally, this triangulation is unique. One of its properties is that the outcircle of every triangle does not contain any other data point. It is used, for instance, when one wishes to construct an approximation to a function z(x,y) whose values are only known for a finite set of points (x,y) (e.g. these could be depth measurements in a canal and the approximation z(x,y) would be used to determine how much sediment has accumulated since an earlier measurement).
This cache is located in a really neat little park we found the other day. It is a nice short walk out with many different ways to get there. Love the green belts.
Please note that not all GPS Receivers will give the same level of accuracy. Dont post revised coordinates.
It's close enough to add excitement to the hunt.
Question: What is a Voronoi diagram? Answer: A Voronoi diagram is a partition of a plane into regions based on the closest point in a given set of points. | 677.169 | 1 |
Pre-Calculus 11 two If DF=4 we have a right triangle, with EF=4√3 = 6.92 Moving F a bit toward D will enable us to make EF a bit longer (7.00) Moving F a bit away from D will allow for EF a bit longer (7.00)
Thursday, May 16, 2013 at 7:11pm by Steve
math If the first bit (left most bit) is a 0, then it can be filled in only each of the bit string can be filled in one way and when bit strings of length eight end with the two bits 00. Each of the remaining five position is represent in the bit string can be filled in 2 ways (i.e...
Friday, September 7, 2007 at 12:59pm by Touseef Ahmed
Calculus / a bit of physics explain why a scalar equation of the line exists in 2-D space, but not in 3-D space.
Thursday, May 17, 2012 at 9:19am by j
Physics Yes I see what you're saying after drawing the diagram...So for the first bit we just use plain substitution of GPE formulas and second bit is trig. =) thanks that seemed easier than i thought.
Tuesday, November 17, 2009 at 6:23pm by Leah
Discrete Math What ...
Sunday, October 30, 2011 at 10:40pm by Tommy
Physics A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.20x10^4 rad/s to an angular speed of 3.14x10^4 rad/s. In the process, the bit turns through 1.92x10^4 rad. Assuming a constant angular acceleration, how long would it take the bit to reach ...
Tuesday, October 23, 2012 at 8:38am by Sam
physics (the algebra bit) I don't know what to do with the m2 a I know I have to do somehthing with that but what I'm not sure adding it to both sides and dividing by m2 gives me a + a or more simply 2a but then I would have to divide by 2 which isn't in the answer my text book gives me
Thursday, July 30, 2009 at 4:21pm by physics (the algebra bit) by Nikki
Calculus Thank you, and I was a bit confused. Thanks though!
Tuesday, July 17, 2012 at 3:56pm by Tara
Applied Calculus I ...
Thursday, March 21, 2013 at 1:33am by Jacob
Physics A...
Monday, October 22, 2012 at 1:34pm by Sam
Question: What is the final angular speed (ω₂) of the drill bit in rad/s? Answer: 3.14x10^4 rad/s
Question: What is the angular acceleration (α) of the drill bit in rad/s²? Answer: (3.14x10^4 - 1.20x10^4) / (1.92x10^4) = 0.5 rad/s² | 677.169 | 1 |
Ratios in Right Triangles
In this lesson our instructor talks about ratios in right triangles. First she talks about trigonometric ratios of sine, cosine, and tangent. Then she discusses trig function, and inverse trig functions. She finishes with a lecture on SOHCAHTOA. Four extra example videos round up this lesson.
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Ratios in Right Triangles
Trigonometry: The study of involving triangle measurement
sine (sin) = opposite/hypotenuse
cosine (cos) = adjacent/hypotenuse
tangent (tan) = opposite/adjacent
SOHCAHTOA
Ratios in Right Triangles
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Question: How many extra example videos are provided to supplement this lesson? Answer: Four | 677.169 | 1 |
Two contemporary proofs can be considered the oldest record of the Pythagorean theorem: one to be found in Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C., see image below), the other in the Euclid's Elements.
The theorem
A right triangle is a triangle with one right angle; the legs are the two sides that make up the right angle, and the hypotenuse is the third side opposite the right angle. In the picture below, a and b are the legs of a right triangle, and c is the hypotenuse:
Pythagoras perceived the theorem in this geometric fashion, as a statement about areas of squares:
The sum of the areas of the blue and red squares is equal to the area of the purple square.
Similarly, the Indian Sulbasutra texts state that:
The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together.
Using algebra, one can reformulate the theorem into its modern expression by noting that the area of a square is the square (second power) of the length of its side:
Given a right triangle with legs of lengths a and b and hypotenuse of length c, then a2 + b2 = c2.
A visual proof
Perhaps this theorem has a greater variety of different known proofs than any other (the law of quadratic reciprocity may also be a contender for that distinction).
This illustration depicts one of them. The area of each large square is (a + b)2. In both, the area of four identical triangles is removed. The remaining areas, a2 + b2 and c2, are equal. Q.E.D.
NB: This proof is very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link).
The converse
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
This converse also appears in Euclid's Elements. This can be proven using the law of cosines which is a generalization of the Pythagorean theorem applying to all (Euclidean) triangles, not just right-angled ones. This generalisation is: a2 + b2 + 2ab cos(ß)= c2. ß is here the angle between sides a and b.
Generalizations
Question: What is the modern algebraic expression of the Pythagorean theorem? Answer: a² + b² = c²
Question: What are the lengths of the sides of a right triangle called? Answer: The legs are called 'a' and 'b', and the hypotenuse is called 'c'
Question: Which part of the illustration shows the area of the square on the hypotenuse? Answer: The purple square
Question: Which theorem is a generalization of the Pythagorean theorem? Answer: The law of cosines | 677.169 | 1 |
If one erects similar figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (x0, y0) and (x1, y1) are points in the plane, then the distance between them is given by
The generalisation of this result to infinite-dimensional inner product spaces is known as Parseval's identity.
The Pythagorean theorem also generalises to higher-dimensional simplexes. If a tetrahedron has a right angle corner (a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This is called de Gua's theorem.
where α is the angle opposite to side a, β is the angle opposite to side b and γ is the angle opposite to side c.
This formula holds in all triangles, not just the right triangles. If γ is a right angle (γ equals π/2 radians or 90°), then sgn(α + β - γ) = 0 since the sum of the angles of a triangle is π radians (or 180°). Thus, a2 + b2 - c2 = 0.
The Pythagorean theorem in non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to π/2; this violates the Euclidean Pythagorean theorem because (π/2)2 + (π/2)2 ≠ (π/2)2.
This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider -- spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:
For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form
By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.
where cosh is the hyperbolic cosine. By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.
Heraldry
In heraldry the Pythagorean theorem appears as a charge in the arms of Seissenegger
Question: What is the relationship between the areas of the smaller and larger right triangles when similar figures are erected on their sides? Answer: The sum of the areas of the two smaller triangles equals the area of the larger one.
Question: What is the form of the Pythagorean theorem in hyperbolic plane geometry? Answer: a² + b² - c² = 2ab cosh(c/a)
Question: In spherical geometry, what is the length of all three sides of the right triangle bounding an octant of the unit sphere? Answer: All three sides have length equal to π/2.
Question: In Euclidean geometry, what is the formula for the distance between two points (x0, y0) and (x1, y1) in the plane? Answer: The distance is given by √[(x1-x0)² + (y1-y0)²]. | 677.169 | 1 |
Two Vectors
ScrawnySpatula8167 asked
The two vectors a and b in the figure have equal magnitude of 7
m and the angles are ?1 = 31° and ?2 = 99°.
a. Find the x-component of their vector sum r. b.Find the
y-component of their vector sum r. c. What is the magnitude of
their vector sum r? d. Find the angle that their vector sum r makes
with the positive direction of the x-axis.
Answers (1)
Answered by Anonymous4 minutes later
Rating:2 Stars
This question can't be answered for you by someone else without
seeing the diagram, or at least giving a better description of it.
The sum vector depends on whether they go in similar directions or
opposite directions.
However, if you follow these steps, you may be able to find the
answer on your own. First of all, if you are going to find the sum
vector, you can copy the second vector onto the end of the first
vector, so that you can follow the first one to the end, then
follow the second one, and see what the result is. If you can
successfully graph this, you can find the coordinates of the
endpoints of r, and also its magnitude and angle.
Question: Can you find the x-component of their vector sum r without a diagram? Answer: No, you need the diagram or a better description to determine the direction of the vectors. | 677.169 | 1 |
A good answer might be:
a = (0,4)T is aligned with the Y axis.
Its length, 4 units, can be read off the diagram.
Pythagorean Formula
Of course, vectors have no fixed location, so vector a
can be drawn anywhere.
The diagram shows the vectors of length 3 and 4, and with a new
vector, h.
The length of vector h
is harder to figure out.
But not much harder, especially if you know about "3, 4, 5 right triangles."
The three vectors can be arranged into a right triangle with
h as the
hypotenuse.
The other sides are 3 and 4,
so the length of h is 5.
You probably know the Pythagorean Formula:
(length of hypoteneuse)2 =
(length of first side)2 + (length of second side)2
Using this with a right triangle with sides of 3 and 4:
(length of hypoteneuse)2 = 32 + 42
(length of hypoteneuse)2 = 9 + 16 = 25
(length of hypoteneuse) = 5
QUESTION 3:
What is the length of the hypoteneuse of a right triangle
whose two sides are 6 and 8?
Question: Is vector a aligned with the X axis? Answer: No, vector a is aligned with the Y axis. | 677.169 | 1 |
Cartesian coordinates are a central part of high school algebra, and the theory continues to have useful applications in geometry ... Polar graphs offer a more natural way of plotting and viewing certain types of data than Cartesian graphs. ...
Representing Data Graphically There is an old saying that "a picture is like a thousand words." This can also be true in mathematics. A graph (mathematical picture) can be more meaningful than a stack of numbers.
Introduction to the Cartesian Plane and plotting coordinates. ... Often, we draw a set of axes on graph paper as shown below. The position of any point on the Cartesian plane is described by using two numbers, (x, y), that are called coordinates. The first ...
Cartesian coordinates, also called rectangular coordinates, provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface or in three-dimensional (3D) space
Question: In which dimension(s) can Cartesian coordinates be used to indicate the positions of points? Answer: Two-dimensional (2D) surface or in three-dimensional (3D) space | 677.169 | 1 |
Hello Friends, I would like you to help me with coordinate geometry. This is a problem from MGMAT. The line is represented by equation y = x is the perpendicular bisector of line segment AB. If AB has the coordinates (-3,3), what are the coordinates of B?From the equations, you can see that the line y=-x is bisected at the origin by y = x. So the midpoint of segment AB is (0,0).
Midpoint of a line segment having points A(x_1,y_1) and B(x_2,y_2) is (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})
We have the values of the point A and the midpoint. So point be can be easily found.
Question: What are the coordinates of point A? Answer: (-3, 3) | 677.169 | 1 |
Tags
180 Rule Demonstration
Activity Overview / Details
Before we begin this section of the lesson, place the students
into groups of four. These will be the production groups for
the video assignment. Hand out the document titled, "Don't
Cross The Line! to each student.
TEACHER* Ask the students to read the hand out, as you bring out
a camera on a tripod.
Now, you need one of the groups to volunteer (or choose one
group) for the technique demonstration.
Ask the class to refer to the handout as the demonstration takes
place.
(I have included a Power Point presentation I use when I
demonstrate this technique. I control the PPT, and have the student
group move the camera around in various locations - following the
180 degree rule.)
PROCEDURE: Have two students stand or sit and pretend to
engage in a conversation.
Ask the other two students to handle the camera and tripod.
STEP 1: Ask the students the following question: "remember
in math class, geometry to be specific - how many degrees are there
in a circle? or How many degrees make up a complete circle?"
ANSWER: 360 degrees.
OK, imagine a circle around the subjects of this scene....
where is the action line, or the 180 degree line? Make
sure to point out the line. (you could use string to
emphasize this line)
Now simply follow along to place the camera along different
angles - all on the same side of the 180 degree line as indicated
from the videos and handout.
Question: How many degrees make up a complete circle? Answer: 360 degrees
Question: What should the camera's position be in relation to the 180 degree line? Answer: On the same side of the 180 degree line as the subjects. | 677.169 | 1 |
Videos, worksheets, games and acivities to help Geometry students learn geometry proofs and how to use CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction.
CPCTC
CPCTC is an acronym for corresponding parts of congruent triangles are congruent. CPCTC is commonly used at or near the end of a proof which asks the student to show that two angles or two sides are congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be congruent.
How to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC.
How do we Use CPCTC?
Two Column Proofs
Two
How to organize a two column proof.
A brief lesson and practice on drawing diagrams and completing two column proofs from word problems
Flowchart Proofs
Flowchart proofs are organized with boxes and arrows; each "statement" is inside the box and each "reason" is underneath each box. Each statement in a proof allows another subsequent statement to be made. In flowchart proofs, this progression is shown through arrows. Flowchart proofs are useful because it allows the reader to see how each statement leads to the conclusion.
Question: What does the acronym CPCTC stand for? Answer: Corresponding parts of congruent triangles are congruent. | 677.169 | 1 |
Do the multiplication by -1 to get:
.
.
But don't forget that you have to reverse the direction of the inequality sign too. When
you do the inequality is now:
.
.
Now you can replace the inequality sign with an equal sign and solve the equation for y
just as you have always done. Begin by adding 4x to both sides to get:
.
.
Divide both sides by 6 to solve for y and get:
.
.
which becomes:
.
.
Graph this equation as you did previously. The slope is (2/3) and the y-axis intercept
is -2.
.
Now replace the = sign with the inequality sign pointing to the right so that the inequality
is now:
.
.
Again, shade the entire region ABOVE the graph of the right side of the equation.
That represents the place where y is allowed. y can NOT be on or below the graphed line.
.
After a little more practice you'll get familiar with this method and you can do things
faster and without thinking about it.
.
Hope this helps you with understanding the basic principles of doing problems such as
these inequalities.
Triangles/70288: This question is from textbook
i have to draw two different triangles that are not congruent, each with angles measuring 50 degrees and 70 degrees 1 solutions Answer 50121 by bucky(2189) on 2007-02-11 17:52:27 (Show Source):
You can put this solution on YOUR website! If two triangles are congruent they will both contain the same angles, and the corresponding
sides will be equal in length.
.
All you need to do to answer this question is to use the same angles in both triangles
but make one of the triangles have longer sides. The triangles will be the same shape but
one will be an enlargement of the other.
.
The way to do this is to draw two horizontal lines on a piece of paper, but make one of the
lines obviously longer than the other. Put angles of 50 degrees at the left end of each of
the two lines you drew. Make the line forming the angle slant up and to the right.
.
Next go to the other ends of the two lines and make 70 degree angles of so the lines
forming the angle slope up and to the left. You should now have triangles that both contain
the same angles, but the corresponding sides of the two triangles are different in length.
.
Hope this helps you understand the problem.
You can put this solution on YOUR website! Two equations are required.
You know that the perimeter of a rectangle is given by adding a length, a width, another length,
and another width to get the perimeter.
Let L represent the Length, W represent the Width, and P represent the Perimeter of the
rectangle. With this notation, the equation for the perimeter of the rectangle is:
.
.
Question: What is the value of y after solving the equation? Answer: y = -3 | 677.169 | 1 |
Given triangle ABC with AB = AC, extend segment AB to a point P so that B is between A and P and BP = BC. In the resulting triangle APC, show that angle ACP is exactly three times the size of angle APC. (By the way, notice that extending segment AB does NOT mean the same thing as extending segment BA.)
thnx!!! 1 solutions Answer 107653 by vleith(2825) on 2008-07-05 21:26:29 (Show Source):
You can put this solution on YOUR website! Let the angle at BAC = a.
Triangle ABC is isosceles, so its base angles are equal. Call those angles b.
We have a+b+b = 180
By extending B to P, with a length equal to BC, we create a second isosceles triangle BPC. The angle at PBC is the supplement of the angle CBA. That angle is b (from above). So angle PBC = 180-b
Since the second triangle also has base angles equal, we know that angle BPC and BCP are the same. Call that angle c.
Then PBC + 2c = 180.
PBC = 180 - 2c
From above PBC = 180-b, so
180-b = 180 - 2c
b/2 = c
The angle at CPB = c = b/2
The angle PCA = c + b = b/2 + b = 3b/2
Which is 3 times CPB
You can put this solution on YOUR website! See this -->
Then substitute in 'numbers' or other combinations of letters to create examples.
For instance:
Transitive Property of Equality: if a = b and b = c, then a = c
example If 12 = (9+3) and (9+3) = (13-1) then 12 = (13-1) OR
if x = y and y = 2z then x = 2z
You can put this solution on YOUR website! I know of 6. Perhaps there are more
1) solving by substitution
2) solving by elimination
3) solving by completing the square
4) solving graphically
5) solving using the quadratic equation
6) factoring
See here for some more words -->
You can put this solution on YOUR website! Not sure what the original equation really is. Look here for a tool -->
test/146158: I am going to choose 6 students out of the 15 who currently have A's to be mentors for the students who have C's or lower. How many ways can a group of 6 be selected? 1 solutions Answer 106712 by vleith(2825) on 2008-06-23 18:36:02 (Show Source):
Question: What are the measures of the base angles of triangle ABC? Answer: They are both 'b'. | 677.169 | 1 |
Pythagorean Theorem
This week in geometry we learned about the Pythagorean theorem. One would use the Pythagorean theorem to find the lengths of the sides of a right triangle. The formula is: leg1 squared + leg2 squared = hypotenuse squared. The hypotenuse is the side opposite the right angle. There are two special triangles when using the Pythagorean theorem. There is the 45 degree-45 degree right triangle, and there is the 30 degree-60 degree right triangle. In the 45-45 right triangle, the hypotenuse = the leg times root 2. In the 30-60 right triangle, the hypotenuse = the short leg times 2, and the long leg = the short leg times root three.
Question: In a 45-45-90 triangle, what is the relationship between the hypotenuse and one of its legs? Answer: The hypotenuse is the leg times √2 | 677.169 | 1 |
String theory
I have deduced an equation of a right triangle using the trigonometric ratio ...
Posts 1 - 2 of 2
String theory discussion
midnightroyale 08/25/10
I have deduced an equation of a right triangle using the trigonometric ratio sine can i apply that equation for example to the moon and two stars to determine the distance from one another?
yuvrajhanspal 09/06/10
replied to: midnightroyale
Hi,
I think not, as the stars are not in the same plane. One of the two could be actually closer to the earth than the other, so what you might end up is the arc distance between those two. I m not sure, but this is my understanding. If you could actually, it would be great :). Try the formula for alpha centauri and sirius. That might help if u get the distance between the two correctly using your formula and the actual distance between the two.
Yuvraj
Question: If midnightroyale's formula calculates the same distance as the actual distance between Alpha Centauri and Sirius, what does yuvraj suggest? Answer: Try the formula for Alpha Centauri and Sirius | 677.169 | 1 |
I assume you're trying to find an angle between two points? I'm sure there are more efficient ways of doing this but I used some stuff I learned from Calculus to develop a Get_Angle() proc which uses inverse cosine and some vector math to determine the angle between point (x0,y0) and point (x1,y1).
atan2() (or arctan2() or arctangent2() or whatever you want to call it) is different from just the arctangent function. atan() takes just one argument, which is the ratio y/x. However, the issue here is that the domain and range is then limited to half of the unit circle: y/x = -y/-x, and -y/x = y/-x. The range of the function, therefore, is only -pi/2 to pi/2. The atan2() function solves this by taking two arguments - y and x - and then using that to give a full range.
Granted, it's only a small difference in terms of lines of code (as shown by Lummox's example) but it really makes things neater and less prone to error.
Also that's a really poor explanation in retrospect. Anybody who's actually a mathematician able to say things properly?
Question: Which mathematical operation does Get_Angle() use to determine the angle? Answer: Inverse cosine (arccos) | 677.169 | 1 |
Area of Kite = $\frac{d_1 d_2}{2}$ where $d_1$ and $d_2$ are the lengths of the diagonals.
Question 4: One side of a kite is 5 cm less than 7 times the length of another. If
the perimeter is 86 cm, find the length of each side of the kite. Solution:
Let the two unequal sides of a kite be a and b. Then we have b = 7a – 5. Perimeter of a Kite = 2a + 2b, where a and b are the lengths of each side in each pair of equal sides. The Perimeter of the given kite = 86 cm. Thus, 86 = 2a + 2(7a - 5) 86 = 2a + 14a - 10 = 16a - 10. 16a = 96
a = $\frac{96}{16}$ = 6 cm.
b = 7a – 5 = 7(6) – 5 = 42 – 5 – 37 cms.
The length of each side of the kite is 6 cm, 6 cm, 37cm and 37 cm.
Question 5: In the figure below if AD = CD, $\angle$CDB = $\angle$ADB, Prove that ABCD is a kite.
Solution:
Statements
Reasons
1. AD ≅ CD
1. Given.
2. ∠CDB ≅ ∠ADB
2.Given.
3. BD ≅ BD
3.Reflexive Property.
4. ΔBCD ≅ Δ BAD
4.By SAS postulate.
5. BC ≅BA
5.By CPCTC.
6. ABCD is a Kite
6.If a quadrilateral has two disjoint adjacent sides that are congruent, then the quadrilateral is a kite.
Question: What is the area of a kite with diagonal lengths of 8 cm and 12 cm? Answer: 48 cm² (calculated as $\frac{8 \times 12}{2}$)
Question: If the length of one diagonal of a kite is 10 cm, what is the length of the other diagonal? Answer: 10 cm (since the diagonals of a kite are equal) | 677.169 | 1 |
Perimeter questions on the GMAT are a subset of a subset — a small part of the geometry you will inevitably run into on the Quantitative side of the test. Perimeter is the distance around a geometric figure — from Ancient Greek περίμετρος, or "measure around". And on the GMAT it is just that: the sum of the side lengths of a two-dimensional figure (three dimensional figures have surface area instead). They are in the category of "easy things made hard". It is not challenging to add up the side lengths of a polygon, even an irregular one, so the GMAT will force you to infer more information to answer the question.
This is a short series of articles showing some of the ways different figures' information can be used to create perimeter questions.
Last time, I gave you this question about triangles:
In the figure above, triangle CDE is an equilateral triangle with side length 3, AF = 10, EF = 6, and AB = DE. What is the perimeter of the figure?
(A) 45
(B) 36
(C) 29
(D) 24
(E) 16
The answer is (C) 29. How?
The question tells us this much:
From here, we can deduce that:
1. Triangle AEF is a 3:4:5 triangle, with side lengths 6:8:10, respectively.
2. Since AE is 8 and CE is 3, that makes AC = 5.
3. Since AC = 5, triangle ABC is also a 3:4:5 triangle, and BC = 4
We can then have:
10+6+4+(3*3) = 29. Answer choice (A) 45 is the sum of the perimeters of each of the triangles, rather than the perimeter of the figure.
Jim - i did not understand what the perimeter of the figure includes. When i calculated, i fell in the trap of adding up all the triangles together (45) which is not the case. Can you explain please what components you took in order to calculate the perimeter? Why did not calculate or add in the lengths of AB and AC ?
Thanks
Question: What is the perimeter of the figure in the given problem, according to the answer provided? Answer: 29 | 677.169 | 1 |
Algebraic curve : A curve whose cartesian equation can be expressed in terms of powers of x and y together with the operations of addition, subtraction, multiplication and division.
For example the astroid, x2/3 + y2/3 = a2/3, is an algebraic curve. The term is due to Leibniz.
Anallagmatic curve : A curve which is invariant under inversion.
The property was first discussed by Moutard in 1864.
Asymptote : A line which is the limit of the tangent to a curve as the point of contact of the tangent tends to infinity.
Bipolar coordinates : Let O and O' be two fixed points. A point P may be specified by giving its distances r and r' from O and O' respectively. These are called the bipolar coordinates of P. A curve may be defined by an equation, called the bipolar equation, connecting r and r'.
For example an ellipse is defined by r + r' = 2a.
Brachistochrone curve : A curve along which a particle will move from one point to another under the action of an accelerating force in the least possible time.
In 1696 Johann Bernoulli put out a challenge to find such a curve where the accelerating force is gravity.
Cissoid : Given two curves C1 and C2 and a fixed point O, let a line from O cut C1 at Q and C2 at R. Then the cissoid is the locus of a point P such that OP = QR.
The cissoid of Diocles is a cissoid where C1 is a circle, C2 is a tangent to C1 and P is the point on C1 diametrically opposite the point of contact of the tangent.
Conchoid : Let C be a curve and O a fixed point. Let P and P' be points on a line from O to C meeting it at Q where P'Q = QP = k, where k is a given constant.
If C is a circle and O is on C then the conchoid is a limacon, while in the special case that k is the diameter of C, then the conchoid is a cardioid.
Curvature : Let C be a curve and let P be a point on C. Let N be the normal at P and let O be the point on N which is the limit of where the normal to C at P' intersects N as P' tends to P. O is the centre of curvature at P and PO is the radius of curvature at that point.
Cusp : A point on a curve C where the gradient of the tangent to C has a discontinuity.
Envelope : A curve which touches every member of a family of curves or lines.
For example the axes are the envelope of the system of circles (x-a)2 + (y-a)2 = a2.
Evolute : The envelope of the normals to a given curve.
This can also be thought of as the locus of the centres of curvature.
Question: What is the radius of curvature at a point on a curve? Answer: The distance from the point to the center of curvature
Question: What is the locus of a point P in a cissoid of Diocles? Answer: The point on a circle diametrically opposite the point of contact of a tangent | 677.169 | 1 |
Pedal curve : Given a curve C then the pedal curve of C with respect to a fixed point O (called the pedal point) is the locus of the point P of intersection of the perpendicular from O to a tangent to C.
Radial curve : Let C be a curve and let O be a fixed point. Let P be on C and let Q be the centre of curvature at P. Let P1 be the point with P1O a line segment parallel and of equal length to PQ. Then the curve traced by P1 is the radial curve of C.
It was studied by Robert Tucker in 1864.
The radial of a cycloid is a circle.
Roulette : Let C1 be a curve and C2 a second curve. Then if P is a point on C2, a roulette is the curve traced out by P as C2 rolls on C1.
A cycloid is the roulette of a point on a circle rolling along a straight line.
Epicycloids, hypocycloids, epitrochoids and hypotrochoids are all roulettes of a circle rolling on another circle.
Spiral : The locus of a point P which winds around a fixed point O (called the pole) in such a way that OP is monotonically decreasing.
Sinusoidal spirals are not true spirals.
Strophoid : Let C be a curve, let O be a fixed point called the pole and let O' be a second fixed point. Let P and P' be points on a line through O meeting C at Q such that P'Q = QP = QO'. The locus of P and P' is called the strophoid of C with respect to the pole O and fixed point O'.
A right strophoid is the strophoid of a line L with pole O not on L and fixed point O' being the point where the perpendicular from O to L cuts L.
Tautochrone : A curve down which a particle acted on by a force will traverse the distance to the lowest point in the curve in a fixed time independent of the starting position.
Transcendental curve : A curve of the form f(x,y) = 0 where f(x,y) is not a polynomial in x and y.
For example the cycloid is a transcendental curve.
The term is due to Leibniz.
Question: Who studied the radial curve? Answer: Robert Tucker, in 1864.
Question: Is a cycloid a transcendental curve? Answer: Yes. | 677.169 | 1 |
Homework 10/5/99
Chapter 3:
Q6. When two vectors are added, the magnitude of the sum will be the greatest
when the vectors point in the same direction. In this case the magnitude will be 7.5
km. When the vectors point in different directions the sum will be smaller.
The sum will be the smallest when the vectors point in exactly opposite directions.
In this case the magnitude of the sum will be 0.5 km.
Q7. Two vectors must have the same magnitude and point in opposite directions to
exactly cancel, that is, to add up to zero. Three unequal vectors can add up to zero
if the two shorter vectors are opposite the longest vector and the sum is zero.
Three unequal vectors can also add up to zero when they form a triangle and the
head of the third vector meets the tail of the first vector
Q8. The magnitude of a vector can equal one of its components when the vector
lies along the x or y axis. If a vector is at an angle to the x or y axis, the
component is smaller than the magnitude of the vector. The component can never be
greater than the magnitude of the vector.
Q9. According to the Pythagorean theorem, the sum of the squares of the
components equals the magnitude squared. When the magnitude squared is zero, all of
the components must be zero since they are all positive when squared and cannot cancel one
another out.
Question: What is the smallest possible magnitude of the sum of two vectors pointing in exactly opposite directions, given that each vector has a magnitude of 7.5 km? Answer: 0.5 km
Question: If the magnitude squared of a vector is zero, what must be true about its components? Answer: All of the components must be zero | 677.169 | 1 |
Vector Functions
Recall that functions are much like computers or machines that take in one or several input numbers and put out a single number. And recall that vectors are mathematical entities composed of two pieces, magnitude and direction, like the...
Please purchase the full module to see the rest of this course
Purchase the Points, Vectors, and Functions Pass and get full access to this Calculus chapter. No limits found here.
Question: Which of the following is NOT a component of a vector? A) Length B) Direction C) Value D) Color Answer: D) Color | 677.169 | 1 |
For a trade to be fair, it has to be fair in both directions. The source producers need to be paid a fair price for the product the produce, but I too need to pay a fair price for the item I'm purchasing.
It's called Pythagoras Theory because it's only a theory. Every triangle that it's ever been tested on works, so there's no reason to doubt that it will continue to work on every triangle we ever find. But until there's a way to prove it, it will always be called a theory. If it's ever proven, it will be called Pythagoras Rule.
Question: Who is the person associated with the mathematical concept? Answer: Pythagoras. | 677.169 | 1 |
Define the maximum and minimum radii and as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis a, the eccentricity is given by
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).
Celestial mechanics
In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipse, in Keplerian, i.e., potentials.
Analogous classifications
A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity:
Classification of elements of SL2(R) as elliptic, parabolic, and hyperbolic – and similarly for classification of elements of PSL2(R), the real Möbius transformations.
Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details.
Question: What is the eccentricity of an ellipse on a triaxial ellipsoid called when it's formed by a section containing both the longest and the shortest axes? Answer: The meridional eccentricity.
Question: In celestial mechanics, when is an orbit said to have low eccentricity? Answer: When the apocenter distance is close to the pericenter distance. | 677.169 | 1 |
Okay — so far, so good. We can now calculate length changes. But many of the structural elements in a strain marker not only change their length, but also their orientation. That is to say, they rotate. In order to quantify that, we need to thinking about shear strain. Shear strain (γ), is calculated from angular shear (ψ), a quantity that is measurable in a rock element (again, provided you have a clear idea of what it looked like before deformation).
For any given line in a strain marker, angular shear is defined as the deviation (in degrees) from perpendicular for a line which was originally perpendicular to the line we care about. That is a ridiculous definition at first glance, and it seems to many students quite non-intuitive why one would bother looking for a line originally perpendicular to the line on which we are actually trying to measure the angular shear! I mean, is that abstractified or what?
Before I reveal my stunning new analogy for angular shear, let me diagram the definition for you, and relate it to shear strain.
The line we care about (blue) looks quite the same after deformation as it looked before deformation. So we can't actually measure anything about it directly that has changed. But when we look at a line which we know was originally perpendicular to it (gray/red), we can measure how it has changed.
The angle between the old perpendicular-line and the new 'perpendicular'- line is ψ, the angular shear. By convention, clockwise rotations of the perpendicular-line are given positive values, while counter-clockwise rotations are assigned negative values. In the above example, the rotation is clockwise by 35°, so we note it as "ψ = +35°".
Another way to think about this situation is to consider any point on our perpendicular-line. Here, let's consider point A:
A was originally at the end of the old perpendicular-line (gray), in position Ao. After deformation, it is now at the end of the new 'perpendicular-line' (red), in position Af. You can see that this outlines a right triangle (yellow):
Right triangles are lots of fun, because they allow us to practice trigonometry. You may remember the mnemonic phrase "SOHCAHTOA" from your high-school trigonometry class. Basically, this relates the angle ψ to the lengths of the sides of the right triangle, where S refers to sin(ψ), C refers to cos(ψ), and T refers to tan(ψ). (That's sine, cosine, and tangent, respectively.) So the change in the position of point A, which could be expressed as Δ(A), is equal to the length of the 'opposite' side relative to the length of the 'adjacent' side of the triangle. The length of the adjacent side isn't changing through this deformation, but the length of the opposite side is changing. With a little deformation, it changes a little. With a lot of deformation, the length of that O side increases dramatically.
Question: Which direction is given a positive value for angular shear by convention? Answer: Clockwise rotations
Question: What is the primary quantity used to quantify the rotation of structural elements in a strain marker? Answer: Angular shear (ψ)
Question: What is the change in the position of point A (Δ(A)) equal to? Answer: The length of the 'opposite' side relative to the length of the 'adjacent' side of the right triangle. | 677.169 | 1 |
Algebra Prove by mathematical induction that 3^(3n+1) + 2^(n+1) is divisible by 5
Geometry Lines p and q are parallel and are intersected by transversal r. If angle 1 = 4x degress and angle 2 = 2x+24 degrees, what is the measure of angle 2?
American Government Proportional representation systems like those in Western Europe encourage the formation of smaller parties by enabling parties to: A. win legislative seats even though they do not receive a majority of votes in elections. B. receive campaign funds from government in proportio...
chemistry im so confused about this question Calculate the pH of a solution that has [H3O+] = 5.5 x 10-3.
Question: What is the value of x in angle 2 (2x+24 degrees)? Answer: 18 degrees (since 2x+24 = 180, x = 18/2 = 9). | 677.169 | 1 |
Question 238515: Prove the following statement:
If a triangle has one obtuse angle, then the other two angles are acute. Found 2 solutions by Fombitz, nyc_function:Answer by Fombitz(13828) (Show Source):
You can put this solution on YOUR website! For every triangle,
where A,B,C are the angles.
An obtuse angle is an angle greater than 90 degrees.
Let A be an obtuse angle.
Then A=90+e, where e is a positive value, less than 90.
B and C cannot equal 0.
Then
which means they are both acute.
You can put this solution on YOUR website! An obtuse angle is an angle whose measure is more than 90 degress but less than 180 degrees.
The sum of the measures of the angles of a triangle is 180 degrees.
Given a triangle having one obtuse angle, it follows that the other two angles must be acute or less than 90 degrees because the sum of the angles of a triangle = 180 degrees.
Question: If angle A is obtuse in a triangle, what is the relationship between A and 90 degrees?
Answer: A is greater than 90 degrees | 677.169 | 1 |
The four transformations; reflection, rotation,
translation and enlargement are studied in more detail
in Year 9 and Year 11.
Download these documents for a look at this topic.
Below is a summary of each:
Reflection
When a shape or point is reflected its image is on
the opposite side of a mirror line or axis of symmetry.
The mirror line is halfway between the shape and its image.
The axis of symmetry is often shown by the letter m.
The object and the image are congruent, the same shape
and size.
Length, angle size and shape are said to be invariant for
reflection, which means they do not change.
Examples of reflections -
The diagram shows a reflection in the mirror line m
Rotation
A rotation is a transformation where a point, or an
object, is turned around a fixed point to a new position called
the image.
The object and the image are the same shape
and the same size but in different positions.
When a shape or point is rotated there
is a centre of rotation which remains fixed.
The angle of rotation gives the number
of degrees that the shape is rotated through.
An anti-clockwise rotation is said to be positive.
A clockwise rotation is said to be
negative.
Examples of rotations -
The diagram shows an anti-clockwise quarter
turn rotation about the centre of rotation, O.
Translation
A translation is a transformation
where all points move the same distance and in the same direction.
The object and the image are the same shape and the
same size.
Translations can be represented by vectors.
In general the vector can
represent a translation where x
is the horizontal movement and y is the vertical
movement.
Examples of translations -
The diagram shows a translation of
Enlargement
An enlargement is a transformation where the size
of an object changes.
Examples of enlargements -
The object becomes larger or smaller.
If points on the object and the corresponding points on the
image are joined with a straight line, these straight lines
meet at the centre of enlargement.
The scale factor for an enlargement tells how much
an object has been enlarged by.
A negative scale factor means the object and the image
are on opposite sides of the centre of enlargement.
The diagram shows an enlargement, centre O with a
scale factor of 2.
Download these files for more help with transformations:
An interactive spreadsheet of reflections, rotations,
translations and enlargements. (Excel)
A explanation about Transformations (Powerpoint).
A worksheet on Reflection (Word).
A worksheet on Rotation (Word).
(Windows users, right click and "Save target as..." to save
the files on your computer. When these files have been downloaded to your
computer you will need to use Microsoft Word, Excel and Microsoft PowerPoint
to open them.)
Question: Which of the following transformations involves a change in size? Answer: Enlargement
Question: What does the letter'm' often represent in a reflection? Answer: The axis of symmetry
Question: In a reflection, are the object and its image the same shape and size? Answer: Yes | 677.169 | 1 |
where called the semi-perimeter and bA, bB, and bC are bisectors of angles A, B, and C, respectively. The given formulas are not worth memorizing for if you are given three sides, you can easily solve the length of angle bisectors by using the Cosine and Sine Laws.
Perpendicular Bisector
Perpendicular bisector of the triangle is a perpendicular line that crosses through midpoint of the side of the triangle. The three perpendicular bisectors are worth noting for it intersects at the center of the circumscribing circle of the triangle. The point of intersection is called the circumcenter. The figure below shows the perpendicular bisector through side b.
Question: What is the significance of the three perpendicular bisectors of a triangle? Answer: They intersect at the center of the circumscribing circle of the triangle | 677.169 | 1 |
Parallelogram Vector addition: In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.
Vector addition component method is one way used in adding vectors. Component means 'part 'and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.
Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector's tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below
As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.
Wednesday, March 20
In mathematics the equations are very important. The problems can be converted into equations and then the equations can be solved with the help of multiple methods. The solutions obtained must be checked for their feasibility. This is because all the solutions obtained will not be feasible. Only the solutions that are feasible must be selected, otherwise the answer might go wrong. One must be very careful in selecting the solutions of the equations.
In case of linearity the degree of the given equation is '1'. The linear combination can be formed with the help of an equation. There need to be constants for this process to be performed. It also involves the simple process of addition. Basically it is the formation of an expression. This expression can be formed with the help of constants and the simple addition process.
The term in the expression must be multiplied with a constant and the answers obtained must be added. This will give the required solution. The linear combinations can be very helpful and can have various applications in mathematics.
Question: What does the term 'component' refer to in the context of vector addition? Answer: The coordinates of the point associated with the vector
Question: What is the sum of vectors u=(4,3) and v=(1,4) using the component method? Answer: (5,7) | 677.169 | 1 |
10th Grade Math: Tangent Line Help
Related Subjects
10th Grade Math: Tangent Line
In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point. As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.
In a similar way, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
Many 10th grade math students find tangent line difficult. They feel overwhelmed with tangent line homework, tests and projects. And it is not always easy to find tangent line tutor who is both good and affordable. Now finding tangent line help is easy. For your tangent line homework, tangent line tests, tangent line projects, and tangent line tutoring needs, TuLyn is a one-stop solution. You can master hundreds of math topics by using TuLyn.
At TuLyn, we have over 2000 math video tutorial clips including tangent line videos, tangent line practice word problems, tangent line questions and answers, and tangent line worksheets.
Our tangent line videos replace text-based tutorials in 10th grade math books and give you better, step-by-step explanations of tangent line. Watch each video repeatedly until you understand how to approach tangent line problems and how to solve them.
Tons of video tutorials on tangent line make it easy for you to better understand the concept.
Tons of word problems on tangent line give you all the practice you need.
Tons of printable worksheets on tangent line let you practice what you have learned in your 10th grade math class by watching the video tutorials.
How to do better on tangent line: TuLyn makes tangent line easy for 10th grade math students.
Tenth Grade: Tangent Line Practice Questions
BC is tangent to A at B and to D at C (not
drawn to scale). If AB = 12 , BC = 18, and
DC = 3, find the length of ...
tangent line homework help questions for 10th grade math students.
Question: What is the length of BC in the given tangent line problem? Answer: The length of BC is 18. | 677.169 | 1 |
Dividing Rules:
When working with rules for positive and negative numbers, try and think of weight loss or poker games to help solidify 'what this works'.
Using a number line showing both sides of 0 is very helpful to help develop the understanding of working with positive and negative numbers/integers.
Midpoint Formula
The Midpoint forumla is used when you need the point that is exactly between two other points. The midpoint formula is applied when you need to find a line that bisects a specific line segment. Essentially, the 'middle point' is called the "midpoint".
The Slope Formula
Sometimes called 'Rise over Run'.
The formula for the slope of the straight line going through the points (x1, y1) and (x 2, y 2) is given by:
AREA
A = (h(a + b))/2, in which h is the height, a the longer parallel side, and b the shorter.
Regular pentagon:
A = 1.720a2, in which a is one of the sides.
Regular hexagon:
A = 2.598a2, in which a is one of the sides.
Regular octagon:
A = 4.828a2, in which a is one of the sides.
Circle:
A = πr2, in which π is 3.1416 and r the radius.
VOLUME
Cube:
V = a3, in which a is one of the edges.
Rectangular prism:
V = abc, in which a is the length, b is the width, and c the depth.
Pyramid:
V = (Ah)/3, in which A is the area of the base and h the height.
Cylinder:
V = πr2h, in which π is 3.1416, r the radius of the base, and h the height.
Cone:
V = (πr2h)/3, in which π is 3.1416, r theradius of the base, and h the height.
Sphere:
V = (4πr3)/3, in which π is 3.1416 and r the radius.
TEMPERATURE SCALES
Degrees Fahrenheit to Degrees Celsius:
TC = 5/9 (TF – 32)
Degrees Celsius to Degrees Fahrenheit:
TF = 9/5 TC + 32
Degrees Celsius to Kelvins:
TK = TC + 273.15
Mean and Median
The arithmetic mean, also called the average, of a series of quantities is obtained by finding the sum of the quantities and dividing it by the number of quantities. In the series 1, 3, 5, 18, 19, 20, 25, the mean or average is 13—in other words, 91 divided by 7.
Question: What is the formula to convert Celsius to Kelvin? Answer: TK = TC + 273.15
Question: What is the mean of the numbers 1, 3, 5, 18, 19, 20, 25? Answer: 13 | 677.169 | 1 |
Question 11959: The question in my books reads as follows; "In the following figure, the measure of angle 1 is 9 degrees less than half of the measure of angle 2. Determine the measure of angles 1 and 2." Using a protractor, this would be simple, but the question appears to be asking for an equation. How would I set this equation up to solve it?
~Kate Click here to see answer by rapaljer(4667)
Question 13981: the question was can the measure of a complementary angle equal the exact measure of 1/2 of a supplementary angle. I did this:
1/2(180-x) = 90-x
90 - x/2 = 90-x
x/2 = x
because i got x/2 = x -- I said yes, and needed to explain, but got it wrong. I am just confused and would appreciate any help.
Question 14151: A pratical question: I was having coffee this morning, and a couple who are building a go cart track were asking this: They want to know the slant of one corner. They know how far it is across the track. What else do they need to know to calculate the angle of the track ? I guessed that the distance accross is one dimension needed. One of the angles would be 90 degrees. How would I figure the smaller angle , which would be the slant of the track ? Click here to see answer by longjonsilver(2297)
Question 14476: Does anyone know how to do this?
Q.) Find the measure of each acute interior angle of a regular pentoGRAM.
.
............................................................................
I know that the triangles of a pentogram have two equal sides and and two congruent angles. And I know that if I divide the pentogram into triangles I get 8 triangles so the sum of the measures of the insides of the intire pentogram is 8x180 = 1440
and
all of the little trianles formed by the pentogram will be the same measure.
...............................................................................
******************************************************************************
Maybe this can help, these are the theorems and corollaries being used in this section:
.
.
Theorem 2.5.2 : The sum(S)of the measures of the interior angles of a polygon with (n) sides is given by S=(n-S)*180 (note that n>2 for any polygon)
.
.
Theroem 2.5.1 : The total number of diagonals(D) in a polygone on (n) sides is given by the formula: D= (n)(n-3) divided by 2
.
.
Corollary 2.5.3 : The measure (I) of each interior angle of a regular polygon of (n) sides is: I=(n-2)*180 degrees divided by (n)
.
.
Question: What is the relationship between the measures of angles 1 and 2 in the given figure? Answer: Angle 1 is 9 degrees less than half of the measure of angle 2.
Question: What is the sum of the interior angles of a regular pentagon? Answer: 540 degrees.
Question: What is the measure of each interior angle of a regular pentagon? Answer: 108 degrees.
Question: What are the two dimensions needed to calculate the angle of a slanted track? Answer: The distance across the track and the vertical height of the track. | 677.169 | 1 |
The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of θ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as the graph of the polar functionr.
Different forms of symmetry can be deduced from the equation of a polar function r. If r(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π−θ) = r(θ) it will be symmetric about the vertical (90°/270°) ray, and if r(θ−α°) = r(θ) it will be rotationally symmetric α° counterclockwise about the pole.
Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
Circle
The general equation for a circle with a center at (r0, φ) and radius a is
r^2 - 2 r r_0 cos(theta - varphi) + r_0^2 = a^2.,
This can be simplified in various ways, to conform to more specific cases, such as the equation
r(theta)=a ,
for a circle with a center at the pole and radius a.
Line
Radial lines (those running through the pole) are represented by the equation
theta = varphi ,,
where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r0, φ) has the equation
r(theta) = {r_0}sec(theta-varphi). ,
Polar rose
A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation,
r(theta) = a cos (ktheta + phi_0),
for any constant phi_0 (including 0). If k is an integer, these equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variablea represents the length of the petals of the rose.
Archimedean spiral
The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation
Question: What is the general equation for a circle with a center at (r0, φ) and radius a in polar coordinates? Answer: r^2 - 2 r r0 cos(theta - varphi) + r0^2 = a^2.
Question: How is the angle of elevation of a radial line represented in polar coordinates? Answer: φ, where φ = arctan m and m is the slope of the line in the Cartesian coordinate system. | 677.169 | 1 |
The cylindrical coordinate system is a coordinate system that essentially extends the two-dimensional polar coordinate system by adding a third coordinate measuring the height of a point above the plane, similar to the way in which the Cartesian coordinate system is extended into three dimensions. The third coordinate is usually denoted h, making the three cylindrical coordinates (r, θ, h).
The three cylindrical coordinates can be converted to Cartesian coordinates by
begin{align}
x &= r , costheta
y &= r , sintheta
z &= h.
end{align}
Spherical coordinates
Polar coordinates can also be extended into three dimensions using the coordinates (ρ, φ, θ), where ρ is the distance from the origin, φ is the angle from the z-axis (called the colatitude or zenith and measured from 0 to 180°) and θ is the angle from the x-axis (as in the polar coordinates). This coordinate system, called the spherical coordinate system, is similar to the latitude and longitude system used for Earth, with the origin in the centre of Earth, the latitude δ being the complement of φ, determined by δ = 90° − φ, and the longitude l being measured by l = θ − 180°.
The three spherical coordinates are converted to Cartesian coordinates by
Applications
Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves — such as the Archimedean spiral — whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems — such as those concerned with bodies moving around a central point or with phenomena originating from a central point — are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.
Position and navigation
Polar coordinates are used often in navigation, as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read niner-zero by air traffic control).
Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.
Question: What is the curve for a standard cardioid microphone represented as? Answer: A polar curve at its target design frequency
Question: Which letter is used to denote the height in cylindrical coordinates? Answer: h
Question: What is the angle in spherical coordinates that is measured from the x-axis? Answer: θ
Question: What is the angle in spherical coordinates that corresponds to the complement of φ? Answer: δ (latitude) | 677.169 | 1 |
To celebrate Halloween, last year we discussed what you can do with 1,818 pounds of pumpkin. It was a popular blog post, and it put an awful lot of smiles on peoples' faces. An entire lamina (filled shape) of smiles, in fact. More »
Wolfram|Alpha already contains many extensive collections of mathematical data, including curves, surfaces, graphs, knots, and polyhedra. However, one type of object we had not systematically incorporated until recently was the class of plane geometric figures technically known as laminae:
Most people (including the subset of small people who play with sorting toys such as the one illustrated below) are familiar with a number of laminae. A lamina is simply a bounded (and usually connected) region of the Euclidean plane. In the most general case, it has a surface density function ?(x, y) as a function of x- and y-coordinates, but with ?(x, y) = 1 in the simplest case.
Examples of laminae, some of which are illustrated above, therefore include the disk (i.e., filled circle), equilateral triangle, square, trapezoid, and 5-point star. In the interest of completeness, it might be worth mentioning that laminae are always "filled" objects, so the ambiguity about whether the terms "polygon", "square", etc. refer to closed sets of line segments or those segments plus their interiors does not arise for laminae. More »
Question: What makes laminae always "filled" objects? Answer: They are always bounded and usually connected regions of the Euclidean plane.
Question: Which of the following is a lamina? (a) A set of line segments (b) A filled circle (c) A closed set of line segments Answer: (b) A filled circle. | 677.169 | 1 |
1 Answer
First of all, there is the question of what it means for points and tangent lines to be "given". I will assume that we have Cartesian coordinates on the plane and that all the given objects are given in these coordinates.
The plan will be pretty straightforward: we'll find the equation of our ellipse, and from that we'll find its radii.
It may seem that 4 points together with their tangent lines is way too much information, but it's not. The thing is, if two of the points are opposite to each other (are inversions of each other through the center of the ellipse), then only one of them gives us any meaningful information. The other one basically repeats what the first one is saying. So, out of the 4 given points, let's pick two that are not opposite to each other. We'll only use these two points and completely ignore the other two.
Let's assume the the center of the ellipse is the origin of coordinates (if it's not, then we can just shift it there), and the two points on the ellipse have coordinates $(x_1,y_1)$ and $(x_2,y_2)$. The general equation of an ellipse centered at the origin is like this:
$$
Ax^2 + By^2 + 2Cxy = 1.
$$
So, we need to find $A$, $B$ and $C$. We know two points on the ellipse, which gives us two equations (linear in $A$, $B$ and $C$):
$$
Ax_1^2 + By_1^2 + 2Cx_1y_1 = 1,\\
Ax_2^2 + By_2^2 + 2Cx_2y_2 = 1.\\
$$
There are $3$ unknowns and $2$ equations. Clearly, this is not enough. So, we'll have to use our tangent lines. Suppose that the tangent line to the ellipse at point $(x_1,y_1)$ has direction $(\Delta x, \Delta y)$. Since the line is tangent to the ellipse, we have an equation:
$$
A x_1(\Delta x) + B y_1(\Delta y) + C (x_1 \Delta y + y_1 \Delta x) = 0.
$$
Now we have 3 equations and 3 unknowns. In general, this doesn't mean anything. But in our case it does. We have picked the two points on the ellipse that are not opposite to each other. My geometric sixths sense tells me that these two points and a tangent line at one of them define the ellipse uniquely, therefore the $3$ equations above should have a unique solution $(A,B,C)$. You should probably try and prove this yourself. It seems to be a rather straightforward technical task, and I'm not up to it at this late hour. Or, if you are writing a program, you can just solve the equations programmatically, make your program crash if the solution isn't unique, and hope that it never actually happens ))
Question: How many equations are formed using the two points on the ellipse? Answer: 2 equations
Question: How many points are initially given to find the equation of the ellipse? Answer: 4 points
Question: What is the general equation of an ellipse centered at the origin? Answer: Ax^2 + By^2 + 2Cxy = 1
Question: What is the relationship between the number of unknowns and equations at this point? Answer: There are 3 unknowns (A, B, C) and 2 equations, which is not enough to solve for the unknowns | 677.169 | 1 |
Four Dogs Running
Date: 08/08/2001 at 04:28:09
From: Devon King
Subject: Geometry/algebra
Four dogs are at four corners of a field. Each dog chases the dog to
its right; all four run at the same speed and no acceleration is
assumed. The questions are:
1. Where will they meet?
2. How far will they have run when they meet?
3. How long will it take them to meet?
I can only assume the they will run a perfect arc before meeting in
the middle. If we assume that the length of the field is x, the
distance run will be 0.25 pi x. If we assume that their speed is s
then the time taken will be 0.25 pi x/s. Is this correct, or is my
first assumption wrong?
Date: 08/08/2001 at 12:44:24
From: Doctor Peterson
Subject: Re: Geometry/algebra
Hi, Devon.
Your assumption goes too far; we can't assume something just because
it makes the problem easy to solve. You can hope it will work out that
way, and look for ways to prove it, but in this case that won't work.
The "pursuit curve" is not just an arc of a circle. But in fact, you
don't need to know the shape of the curve.
Properly, this kind of "pursuit problem" belongs in the study of
calculus. Even the trick answer I'm familiar with depends on the sort
of thinking you do in calculus. But here's the basic idea; see if you
can follow the reasoning.
Picture a short interval of time after the dogs start running. Each
dog will have run a short distance almost along the edge of the
square, since in that short time the dog it is chasing will not have
moved much. If you draw this (exaggerating how far they have moved, so
you can see it), you will find the dogs are still at the corners of a
square, but the square is tilted slightly. The distance each dog has
run, plus its distance to the dog it is chasing, is (approximately, if
the time is short enough) the length of a side of the original square.
If you continue this for repeated small intervals, you will see a
sequence of more and more tilted, smaller and smaller squares. But the
sum of the distance a dog has gone, and the distance it has left to
go, will not change, because the dog in front is always moving
perpendicular to the direction your dog is going. See if this gives
you an idea as to how far the dog ran.
It's hard to be fully convincing about this without some pretty deep
thinking; but you can find a fuller discussion of both the calculus
methods (which you can skip) and the trick, in this answer from our
archives to a harder question:
Catching a Pig
Question: Are the dogs running at different speeds? Answer: No, all four dogs run at the same speed
Question: According to Doctor Peterson, is Devon King's assumption about the dogs running a perfect arc correct? Answer: No, Devon King's assumption is incorrect | 677.169 | 1 |
In my prior post about using Clifford algebras to do plane rotations, I finished with a non-intuitive step at the end. Rather than multiplying on the right by an element representing a rotation of angle , I multiplied on the left by an element representing a rotation of angle and multiplied on the right by an element representing a rotation of angle .
Why did I do this? Well, I mentioned it would be awkward for the two-dimensional case, but that it will be important when we get to three or more dimensions. Well, work for a moment with being a quarter rotation (ninety degrees, radians). This means our total rotation is going to be a half turn (180 degrees, radians).
For that , and so . Let's just look at what it does to our unit vectors and to multiply on the left by and on the right by .
For , we get . Similarly, for , we get .
So far, we were only working in two dimensions. As such, there wasn't any to worry about. But, what if there were? What happens to the -coordinate of something if you rotate things parallel to the -plane? It remains unchanged.
Well, what would happen if we multiplied on the right by ? We would end up with . We've ended up scaling and adding in a trivector . We've made a mess.
Let's try it instead with our trick. We're going to start with . Every time we transpose elements with different subscripts, we flip the sign. Every time we get two elements next to each other with the same subscript, they cancel out. So, switching the with the second , we get . From there, we can switch the first two elements to get which is just . We can switch the with the second to get: which is just . So, our trick leaves unchanged.
In the above, there is nothing special about the subscript three. It would work for any subscript except one or two. So, the trick allows us to break the rotation up into two parts that still do what we want with and but leave our other directions unchanged (or, maybe it's easier to think of them as changing them and then changing them right back).
Question: Why was this step awkward in the two-dimensional case but important for three or more dimensions? Answer: The author mentioned that this step would be awkward for the two-dimensional case but will be important when dealing with three or more dimensions.
Question: Which subscripts does the author's trick not work for? Answer: The trick does not work for subscripts one or two. | 677.169 | 1 |
Definition of the Trigonometric Functions of an Acute Angle
posted on: 28 Jun, 2012 | updated on: 29 Jun, 2012
Before going to discuss the definition of the Trigonometric Functions of an Acute Angle let's know about acute angle. An acute angle is an angle which is always lesser than 90 degrees. It does not include 90 degrees anyways. In order to find Trigonometry functions we need an acute angled triangle. And the triangle must be a Right Triangle. The angle which is of 90 degrees is called the Right Angle in a triangle. The opposite side to a right angle is called hypotenuse and rest sides will be called as legs of that triangle.
Here three legs are associated with the angle θ. One is the hypotenuse which is belonging to the angle θ. The second is opposite side and the last is the adjacent side. We denote hypotenuse by r, opposite by b and adjacent side by a.
In a right angle triangle any two sides of right triangle have a Ratio in the form of a relation which is one to one. It helps to form the different Trigonometry formulas and from this we can derive six formulas as the ratio of hypotenuse and opposite, opposite and adjacent, adjacent and hypotenuse and so on.
Now let see the trigonometry function of acute angles θ in the form of ratio of the sides of a right triangle are:
1: sinθ= b/r
2: cosθ= a/r
3: tanθ= b/a
4: cscθ= r/b
5: secθ= r/a
6: cotθ= a/b
Note that sinθ is reciprocal of csc θ, cos θ of sec θ and tan θ of cot θ.
Now see the Pythagorean formula for r, 'a' and 'b'.
Here hypotenuse is always equal to the Square root of the addition of square of opposite side and adjacent side in any right triangle.
r2 = a2 + b2
Just like this opposite and adjacent also can be find by this formula:
a2 = r2 - b2
And
b2= r2 - a2
These are some Trigonometric Functions of acute angles.
Question: What is the formula for the hypotenuse (r) in terms of the opposite side (b) and the adjacent side (a)? Answer: r² = a² + b² | 677.169 | 1 |
On the plane, a figure that we want to call a triangle has all of its angles on the "inside." Also, there is a clear choice for inside on the plane; it is the side that has finite area. See Figure 6.5. But what is the inside of a triangle on a sphere? The restriction that the area on the inside has to be finite doesn't work for the spherical triangles because all areas on a sphere are finite. So what is it about the large triangle that challenges our view of triangle? You might try to resolve the triangle definition problem by specifying that each side must be the shortest geodesic between the endpoints. However, be aware that antipodal points (that is, a pair of points that are at diametrically opposite poles) on a sphere do not have a unique shortest geodesic joining them. On a cylinder we can have a triangle for which all the sides are the shortest possible segments, yet the triangle does not have finite area. Try to find such an example. In addition, a triangle on a cone will always bound one region that has finite area. Look at some of these ornery examples of triangles. A triangle that encircles the cone point may cause problems. Covering spaces can help you in your investigation of these triangles. For example, what happens when we try to unwrap or lift one of these triangles onto a covering space?
Figure 6.5. Insides of a plane triangle.
Problem 6.4. Angle-Side-Angle (ASA)
Are two triangles congruent if one side and the adjacent angles of one are congruent to one side and the adjacent angles of another?
Figure 6.6. ASA.
Suggestions
This problem is similar in many ways to the previous one. As before, look for counterexamples on all surfaces, and if ASA doesn't hold for all triangles, see if it works for small triangles. If you find that you must restrict yourself to small triangles, see if your previous definition of "small" still works; if it doesn't work here, then modify it.
There are also a few things to keep in mind while working on this problem. First, when considering ASA, both of the angles must be on the same side — the interior of the triangle. For example, see Figure 6.7.
Figure 6.7. Angles of a triangle must be on same side.
Let us look at a proof of ASA on the plane:
Figure 6.8. ASA on the plane.
The planar argument for ASA does not work on spheres, cylinders, and cones because, in general, geodesics on these surfaces intersect in more than one point. But can you make the planar work on a hyperbolic plane? (You may want to modify the planar proof to use only reflections.)
Question: What is the challenge posed by large triangles on a sphere regarding our view of a triangle? Answer: The restriction that the area on the inside has to be finite doesn't work for spherical triangles because all areas on a sphere are finite.
Question: What is the "inside" of a triangle on a plane? Answer: The side with finite area.
Question: Can a triangle on a cylinder have all the sides as the shortest possible segments but not have finite area? Answer: Yes. | 677.169 | 1 |
Search Loci: Convergence:
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
Loci: Convergence
Van Schooten's Ruler Constructions
Solution to Problem X
Problem X: Given three straight lines AG, BC and AD, to find a fourth proportional DE, that is so that AB is to BC as AD is to DE.
This problem can be done in the manner of Euclid, putting the first two lines AB, BC in a straight line AC, and the third on another line AE, which forms an angle CAE with AC. If BD is drawn parallel to CE, then DE is the fourth proportional being sought.
Question: What is the problem described in the text as Problem X? Answer: Given three straight lines AG, BC, and AD, the problem is to find a fourth proportional line DE such that AB is to BC as AD is to DE. | 677.169 | 1 |
8.3.1 Connect the measurement of length, liquid, and mass with the appropriatecustomary and Metric units.
8.3.2 Convert units within and between measurement systems using conversionratios. Convert between various units of area and various units of volume. Calculate and convert rates through conversion ratios and label correctly.
8.4.1 Describe, classify, and compare geometric figures and solids.
8.4.2 Find the missing sides of similar figures using proportions. Find the missing sides of right triangles using the Pythagorean Theorem.
12.4.3 Classify and compare two- and three-dimensional shapes in terms of congruence and similarity and apply the relationships. Apply properties, draw images, and make conclusions of and about transformations (translations, reflections, rotations, and glides).
12.4.4 Calculate distance and midpoints between two points on a coordinate graph. Draw right triangles between any two points and apply the Pythagorean Theorem.
12.4.5 Know and use the definitions of sine, cosine, and tangent. Estimate values of trigonometric ratios. Use sin, cos, and tan to determine unknown lengths in real situations.
12.4.6 Find missing angles, arcs, chords made by the intersection of secants, tangents and circles. Determine measurements of angles formed by parallel, perpendicular and transversal lines. Use the Polygonal Sum Theory and the Pythagorean Theorem.
12.4.4 Transform an equation of a conic section (general form to/from vertex form) and graph.
12.4.5 Approximate and give exact values of trigonometric functions with and without a calculator. Determine the measure of an angle given its sine, cosine, or tangent. Identify and use definitions and theorems relating sines, cosines, and tangents. Solve real-world problems and find missing parts of triangles using right triangle trigonometry, Law of Cosines, and Law of Sines.
12.5.1 Use matrices to store data and to represent and solve real-world situations.
12.5.2 Fit an appropriate model to data (linear, quadratic, power, exponential). Relate the slope (rate of change) and intercepts (initial values) of a regression line. Predict values from a mathematical model (linear, quadratic, power, exponential). Consider the correlation coefficient and determine the validity of predictions made from regression equations.
12.5.4 Identify relationships between figures and their transformation images.
12.6.3 Write, solve and graph linear inequalities in one and two variables.
12.6.4 Identify patterns as sequences (geometric or arithmetic) and write general and recursive definitions. Identify, translate, write, graph and solve variation problems. Determine whether a relation defined by a table, a list of ordered pairs or a simple equation is a function.
12.... Transform equations from vertex form to standard form, and vise versa. Use the discriminant of a quadratic equation to determine the nature of the solutions to the equation.
Question: What is the name of the theorem used to find the measure of an angle given its sine, cosine, or tangent? Answer: The Inverse Trigonometric Theorem.
Question: Which mathematical concept is used to convert units within and between measurement systems? Answer: Conversion ratios.
Question: What is the name of the theorem used to find the missing sides of right triangles? Answer: The Pythagorean Theorem. | 677.169 | 1 |
Question 252434: triangle A is similar to triangle B. side length of triangle A is 3 inches .side length of triangle B is 12 inches. the area of A is 10. what is the area of B ?
a 20 b 40 c 160 d 640 e 40640 Click here to see answer by drk(1908)
Question 253146: Tim says that if the measure of one angle of an isosceles triangle equals 30 degrees, then the measure of the other two angles must each be 75. Marc disagrees he says that while Tim is correct some of the time, there is another possible solution.support Tim's claim and Marc's claim.explain how the angle measures where found please. Click here to see answer by edjones(7569)
Question 253407: i have two questions that i really need help with, help would be very appreciated.
1. A parcel of land is in the shape of an isosceles triangle. the base had a length of 673 feet and the two equal legs meet an angle of 43 degrees. Find, to the nearest square foot, the area of the parcel of land.
i know that area equals one half base times height but i have no idea how to figure out the height i dont know if i should cut the triangle in half or what.
2. The triangle top of a table has two sides of 14 inches and 16 inches and the angle between the sides is 30 degrees. Find the area of the tabletop, in square inches.
i drew i diagram and have attepmted several ways but none of the answers makes sense should i use sohcahtoa i desperatley need help. Click here to see answer by Alan3354(30993 stanbon(57347 scott8148(6628)
Question 250244: I don't have scientific calculator, I need solution for this problem. :-
The measures of the angles of a triangle are (9 square root of 2x + 17 to the power 0) , (9 squareroot x) to the power 0 and (12 squareroot of x + 33 ) to the power 0 . find the measure of each angle? classify the triangle by its angles ? Click here to see answer by richwmiller(9135)
Question: What is the relationship between the angles of an isosceles triangle, given that one angle is 30 degrees? Answer: If one angle is 30 degrees, the other two angles must each be 75 degrees, making the triangle equilateral.
Question: In the text, who is the person who agrees with Tim's claim about the angles of an isosceles triangle? Answer: The person who agrees with Tim's claim is not mentioned in the text.
Question: What is the area of a triangular tabletop with sides of 14 inches and 16 inches and an angle of 30 degrees between them? Answer: The area is 91 square inches. | 677.169 | 1 |
2 The word Sine is of doubtful origin, according to the preamble to Hutton's Mathematical Tables, p.17. It is of some interest to note that the right - hand smaller arc of the first diagram in [Figure 1-1] can be thought of as representing a bow or arcACB, while the string or cord is the chordAB; the length of the bolt or arrow is the sagittaEC. The tangent AB is the line in the lower diagram which touches the circle at A (tangere: to touch), while the secant BC is cut by the circle (secare: to cut) at F. Note in passing that what we now consider as ratios in elementary trigonometry or functions of the angle EXB or in analysis were originally considered as lengths w.r.t. a given radius, which was usually given by a large power of ten. There is no convenient explanation, however, for the use of the word sine for the length of the half chord EA. Hutton considers it to be of Latin origin, in which the word sinus has various related meanings, namely a fold (of the toga at the breast), a hollow, a bay or gulf, etc. This book also provides some useful information on some of the early tabulators mentioned by Briggs in this chapter.
For the half chord EA or EB, Briggs uses the word 'Subtensus, -a, -um' as an adjectival passive past participle, meaning, '(being) stretched or (being) held under' the corresponding arc, where the use of the passive 'being' is optional in English, and does not change the meaning. For convenience, we will always call this the 'subtended chord', or just 'chord' though the word 'chord' is not in the original text of Briggs. Thus, if the Latin text states 'Subtensae AC', we translate this as' of the [Subtending] Chord AC', or 'to/for the Chord AC'.
3 Anyone wishing to know more about the origins of the table of sines can do little better than to spend some time reading Book I of Ptolemy's Almagest. This is readily accessible, e.g. in Volume 16 of The Great Books of the Western World, (Ency. Brit.), where the famous table of Chords is set out on pp.21 - 24, and an explanation given for their construction.
Question: What is the tangent AB in the lower diagram? Answer: The line that touches the circle at A
Question: What word did Briggs use for the half chord EA or EB? Answer: 'Subtensus' | 677.169 | 1 |
typo divide into four right triangles 6, 8 are legs so 3,4,5 triangles, hypotenuse is rhombus side = 10 10*4 = 40
Sunday, January 2, 2011 at 2:31pm by Damon
geometry divide into four right triangles 6, 4 are legs so 3,4,5 triangles, hypotenuse is rhombus side = 10 10*4 = 40
Sunday, January 2, 2011 at 2:31pm by Damon
geometry A geometry example of an if/then statement would be: If two sides and the included angle of a pair of triangles are equal, then the triangles are congruent.
Wednesday, May 28, 2008 at 10:15pm by drwls
math Viviani's Theorem proves this for isosceles triangles. If you take a look at where it is proved for equilateral triangles, you should be able to see that it applies equally well to arbitrary triangles. If not, you...
Sunday, December 2, 2012 at 9:01am by Steve
Geometry. Grade 10. Similar triangles the ratios of the ares of two similar triangles is 4 : 9. What is the ratio of their altitudes drawn from corresponding sides?
Wednesday, December 9, 2009 at 2:00am by John. F
Math A side and the two angles at either end completely define the point where the two other sides intersect. Thus all sides and angles are reproduced for both triangles and the triangles are congruent. That is not the way they want you to wrte geometric proofs, but I hope you can ...
Friday, December 17, 2010 at 10:29pm by drwls
algebra Since the sides are in common ratio between triangles, the triangles are similar. So, corresponding angles are equal. Angle R id congruent to Angle F.
Monday, December 12, 2011 at 4:27pm by Steve
geometry triangles ABF and EDG are congruent. triangles ABF and GCF are equalateral. AG = 24 and CG = 1/5 AB. find the total distance from A to B to C to D to E
Friday, July 15, 2011 at 5:45pm by jordan
Ge 11:15pm by lisa
ge 8:10pm by lisa
math It is a proof. Given: line DB bisects line AC line AD is parallel to line BE AD=BE Prove: DB=EC there are two triangles connected together by point B. They are labled A D B and B E C. D and E are the top points of the triangles. they look like they would be right angle ...
Sunday, October 8, 2006 at 8:52pm by ann
Question: What is the total distance from A to B to C to D to E, given that triangles ABF and EDG are congruent, ABF and GCF are equilateral, AG = 24, and CG = 1/5 AB? Answer: The total distance is 24 + (1/5) * 24 = 36
Question: If two sides and the included angle of a pair of triangles are equal, what can be concluded about the triangles? Answer: The triangles are congruent.
Question: What is the relationship between DB and EC in the proof provided by ann? Answer: DB=EC | 677.169 | 1 |
Generate polygons from a set of intersecting lines As the code description says it won't work as expected by the question author: "edges must be correctly noded; that is, they must only meet at their endpoints. The Polygonizer will run on incorrectly noded input but will not form polygons from non-noded edges"
How to identify a FRONT LAND? To detect the corners I would suggest to use method 1, it would select the whole corner (the front and the side) with that you could then select the longest segment. (as shown in the pictures, corner lands have the front on the longest segment - edit: there is two exceptions on the top center).
Question: Are there any exceptions to the rule that the front is on the longest segment? Answer: Yes, there are two exceptions mentioned at the top center. | 677.169 | 1 |
is directly overhead --altitude equals 90 degrees...
At the pole, the latitude is 90 degrees.
For a different perspective you can focus in on the ship
to see the angular height of Polaris rising congruent with angle oflatitude.
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only. Please cite the contributing author in credits.
All other uses require the express written permission of the respective
contributors.
Question: Where is the latitude 90 degrees? Answer: At the pole | 677.169 | 1 |
It's bh/2, where b=the length of the base (whichever side you are calling the "bottom" of the triangle) and h=the height, or the vertical distance from the line containing the triangle's base to the angle formed by the other two sides. It's a lot easier to show than to explain in words, but I don't have time at the moment to fiddle around with ASCII graphics trying to draw it.
No, that's not it. The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the 2 shorter sides is equal to the square of the length of the hypotenuse. (a^2 + b^2=c^2) BTW, the Scarecrow got it wrong.
In this case, find divide the 6x6x6 triangle down the middle and consider either half, which has a base of 3 and a hypotenuse of 6. The height = sqrt(6^2-3^2) = sqrt(27) = approximately 5.196. That's the height of your triangle.
Well, using the definition of sin, h is equal to the length of the (either one) side with which it has a common vertex multiplied by the angle between this side and the base (on which the h "falls"). So A=a*b*sin(angle between a and b).
You cannot post new topics in this forum You can reply to topics in this forum You cannot edit your posts in this forum You cannot delete your posts in this forum You cannot vote in polls in this forum
Question: What is the height of a 3x6 right triangle? Answer: √(6² - 3²) = √27 ≈ 5.196 units.
Question: What is the length of the hypotenuse of a 3x6 right triangle? Answer: 6 units. | 677.169 | 1 |
The drawing shows a right triangle ABC with a cevian BD. If
angle A = 2x, angle CBD = 3x, and AB = BC + CD, find x. A cevian is a line segment in a triangle with one endpoint on a vertex and the other endpoint on the opposite side.
Geometry problem solving to Solve It, Interactive Mind Map
George Pólya's 1945 book "How to Solve It, A new aspect of Mathematical Method", is a book describing methods of problem solving. It suggests the following steps when solving a mathematical problem:
(1) First, you have to understand the problem.
(2) After understanding, then make a plan.
(3) Carry out the plan.
(4) Look back on your work. How could it be better?
Mind Map Help. To see a note: Hover over a yellow note button. To Fold/Unfold: click a
branch. To Pan: click
and drag the map canvas.
Problem Solving. Activate Flash plugin or Javascript and
reload to view the Mind Map of How to Solve It, Interactive Mind Map, Elearning.
Question: According to George Pólya's steps, what is the first step in solving a mathematical problem? Answer: Understand the problem | 677.169 | 1 |
Coordinate Of A Point
posted on: 25 May, 2012 | updated on: 12 Sep, 2012
A number pair which represents the location of a Point in two - dimensional space is known as coordinates of a point.
Suppose we have Coordinate of a Point R (5, -9) then it defines the location of points.
Where 'R' is the name and the Numbers in the bracket is the 'x' and 'y' coordinates. The first number or x – coordinate denotes how far the horizontal axis is and y- coordinates denote how long up and down the axis is.
The x – axis or horizontal is also known as Abscissa of the points.
The y – axis or vertical horizontal is also known as 'ordinate' of the points.
When we have two pairs in a plane and we want to find the Distance between two points then we can find the distance between two points.
Formula is given for finding the distance between two points is:
D = √ (u2 - u1)2 + (v2 - v1)2;
And in the case of three coordinates (u1, v1, w1) and (u2, v2, w2) the formula for finding the distance is given by:
D = √ (u2 - u1)2 + (v2 - v1)2+ (w2 - w1)2;
Now we will see how to find the distance between the points;
Let the coordinates of points are (-3, -2) and (1, 6) then we have to find the distance between two points.
We know that the coordinates of the points is (u1, v1) and (u2, v2);
Here the value of u1 = -3;
And the value of u2 = 1;
The value of v1 = -2;
The value of v2 = 6;
Then the distance between two points is:
The formula for finding the distance between two points is:
D = √ (u2 - u1)2 + (v2 - v1)2;
Then put the values in the given formula:
D = √ (1 - 5)2 + (6 – (-2))2;
On further solving we get:
D = √ (-4)2 + (8)2;
D = √ 16 + 64;
D = √80;
D = 8.94
So the distance is 8.94.
Using this formula we can find the distance between points.
Topics Covered in Coordinate Of A Point
Before understanding the meaning of abscissa coordinate it is necessary to know about the coordinate plane. A rectangular plane which consists of two number lines i.e. vertical number line and the horizontal number line is known as Coordinate Plane. As we know that horizontal line is represented by the x - axis which is also said to be abscissa and vertical line is known as Ord...Read More
Question: What is another name for the 'y' coordinate? Answer: Ordinate
Question: Which formula is used to find the distance between two points in a plane? Answer: D = √ (u2 - u1)² + (v2 - v1)² | 677.169 | 1 |
math214 A student asks what exactly Euclidean geometry is. How do you answer?
math 214 A student claims that if any two planes that do not intersect are parallel, then any two lines that do not intersect should also be parallel. How do you respond?
math214 A student asks whether a polygon whose sides are congruent is necessarily a regular polygon and whether a polygon with all angles congruent is necessarily a regular polygon. How do you answer?
math214 Maggie claims that to make the measure of an angle greater, you just extend the rays. How do you respond?
math 214 A student wants to know what it means for two planes to be perpendicular. How do you respond?
math 214 Henry claims that a line segment has a finite number of points because it has two endpoints. How do you respond?
Math214 Jane has two tennis serves, a hard serve and a soft serve.Her i...
Language Studies Are those supposed to be considered complete sentences or just an implicature? Thanks!
Language Studies Practice One: Speaker 1: Was there a fiddler at the bar last night? Speaker 2: There was a man scraping a bow across a violin. Implicature:__________________________ Speaker 1: Do you like my new carpet? Speaker 2: The wallpaper's not bad. Implicature:_____________________...
math the following statement always true, always false, or true in some cases and false in others? Explain your answer. A quadrilateral whose diagonals are congruent and perpendicular is a square.
Question: Are the sentences in the Language Studies text complete sentences or implicatures? Answer: The sentences in the Language Studies text are not complete sentences; they are implicatures, which are what is implied by an utterance rather than what is literally said.
Question: Are the student's statements about congruent sides and angles in a polygon true? Answer: Yes, a polygon with all sides congruent is necessarily a regular polygon, but a polygon with all angles congruent is not necessarily regular, as it could be a parallelogram or a rectangle with non-congruent sides. | 677.169 | 1 |
Find the Unit Vector with Positive Component Which is Normal to the Surface?
Unit vector can be defined as a vector which is used to represent the direction of any vector quantity. Length of unit vector is equals to 1. Unit vectors in Cartesian coordinates can be given as
i^, j^ and k^. Here i^ is the unit vector along x – axis, j^ is the unit vector along y – axis and k^ is the unit vector along z – axis. For instance, if force on a body is represented by expression 4i^ then indicates that magnitude of force is 4 N and it is acting along x – axis that is the direction of force is along x – axis. Magnitude of unit vector is always one ('1').
Unit vector of any vector A -> = a1 i^ + a2 j^ + a3 k^ will be given by Ratio of vector to magnitude of vector. This is shown below.
u^ = (A->) / │(A->)│,
Normal vector can be defined as unit vector which is normal to surface for which it is defined.
Let's consider the following diagram for which Normal Vector can be defined. Let's consider the following diagram for which the normal vector can be defined.
Here in above diagram, when vector A-> will be rotated towards vector B->, then according to right hand rule, direction will be normal to the surface. This normal direction is represented by a vector known as normal vector and it is denoted by 'n^'. Here if vector A-> and vector B-> are in X-Y plane then direction of rotation will be along z – axis. Let's derive the formula to find the unit vector with positive component which is normal to the surface.
Here normal vector can be found by cross multiplication of A-> and B-> as shown below:
A->* B->= │A->││* B->│sin x n^
And hence normal vector will be given by
n^ = (A->* B->) / │A->││* B->│sin x.
Question: What is a normal vector? Answer: A normal vector is a unit vector that is perpendicular to a given surface. | 677.169 | 1 |
So in effect, this 23.5º angle from the centre of the King's Chamber intersects a point (the centre of the green cross in Fig. 8) that is already marked-up to the value of 23.5.
Figure 8 - The perpendicular angle of 23.5º that runs through the centre of the KC and the point where the centreline of the GP intersects the floor of the QC - the floor being level with the 23.5th Course Layer
It appears that the same number of Course Layers from the base of the GP to the floor of the Queen's Chamber was used by the architects to verify this angle and its alignment.
One could argue with this and ask why go to all this trouble? Why not centre the two chambers on an angle of 23.5 degrees if this was the original intention?
The simple answer is that for the architects this was impossible given the 51.84º angle of the sides, which appear to have been vital to the overall design.
If the architects had already determined the positions of both chambers via the 23.5º and 6.5º angles to the centre of the KC, and the two 11.75º angles that are half the 23.5º value for the QC, then it would have been impossible aligning the centres of both these chambers on a perpendicular 23.5º angle. The only way this would be possible was if the angles of the sides were 48.42º - a difference of 3.5º. This means that the side angles of 51.84º was of paramount importance.
The geophysical-associated message has already been made clear with the angles we have already discovered, and as we will see later, this perpendicular angle, although important, would have been considered secondary in the overall plan.
So to perhaps show that this alignment would have been ideal in practice and was originally intended for reasons which will soon be made clear, the next best thing was to make sure that the intersection point of both the centreline of the GP and the floor of the QC were connected to the centre of the KC by an angle of 23.5º, and that as an afterthought it was decided that the number of Course Layers from the base to the floor of the QC would both reflect and confirm the value of this intended angle.
This could be argued of course, but this is our explanation as to how the architects may have got around the problem of not being able to centre both chambers on a 23.5-degree angle in addition to the other alignments they had already made and on which the overall geometry of the GP depended.
Question: What is the difference between the original angle of the sides and the angle that would have allowed the chambers to be centered on a 23.5-degree angle? Answer: 3.5 degrees
Question: What is the angle that intersects the center of the King's Chamber and the center of the green cross in Figure 8? Answer: 23.5 degrees | 677.169 | 1 |
Summary: Teaching isometries of the plane plays a major role in the formation of the congruence-concept in the Hungarian curricula. In the present paper I investigate the way the isometries of the plane are traditionally introduced in most of the textbooks, especially the influence of the representations on the congruence concept, created in the teaching process. I am going to publish a second part on this topic about a non-traditional approach (Forming the concept of congruence II). The main idea is to introduce the isometries of the two dimensional plane with the help of concrete, enactive experiences in the three dimensional space, using transparent paper as a legitimate enactive tool for building the concept of geometric motion. I will show that this is both in strict analogy with the axioms of 3-dimensional motion and at the same time close to the children's intuitive concept of congruence.
Question: How does the author propose to introduce isometries in the second part? Answer: The author suggests introducing isometries using concrete, enactive experiences in three-dimensional space, with transparent paper as a tool. | 677.169 | 1 |
Friday, October 3, 2008 at 6:06am by bobpursley
the big idea of energy Copy and paste for the win: The "big idea" of energy, is a phrase designed to point out that everything in the universe is energy in one form to another and that the sum of energy is maintained. Energy can only be transferred or transformed from one form or another ...
Tuesday, December 5, 2006 at 2:18pm by Sam Matthews 11:51pm by John
math in a big right triangle with a small right triangle inside of it. C is the length of the whole thing., R is the adjacent of the small tri. H is the oopisite of the small tri. B is the adjacent of the big tri. as well as the hypotenuse of the small tri. S is the oppisite of the...
Monday, March 21, 2011 at 12:31pm by anonymous
English 1. What nice weather it is! 1-2. What a nice weather it is! (Which one is right?) 3. What a tall dog it is! What...
Tuesday, December 23, 2008 at 2:53pm by John
Question: What is the relationship between the two triangles described in the second entry? Answer: The second entry describes a large right triangle containing a smaller right triangle. | 677.169 | 1 |
Wheels and carts
You must be familiar with wheels and carts.you all must have seen that in your sourrindings as well.so just get into that and explore the world(universe)of circles.
A circle is a closed curve.There are any terms associated with circle.
1.CIRCUMFERENCE = It is the distance around the circle.It is the length of the circle.
2.RADIUS:=It is aline that runs between the circle and its centre.
3.DIAMETER:= It is a chord that goes through centre.Its length is twice the length of radius. d=2*r (where d is the diameter of the circle & 'r'is the radius of same circle.
4.chord:=It is a line segment that join two points on the circumference of circle.
TASKS TO BE DONE[HOMEWORK] 1.DRAW A CIRCLE AND LABEL ITS CENTRE,DIAMETER,CHORD &RADIUS. 2.USE A COMPASS TO MAKE TWO DESIGNS IN TWO SEPARAE CIRCLES. 3.USE A COMPASS,DRAW A CIRCLE OF 3,5 & 7 c.m RESPECTIVELY. Reference
Did you like this resource? Share it with your friends and show your love!
Question: What is the largest radius among the circles to be drawn in the third task? Answer: 7 cm | 677.169 | 1 |
geometry
just started at geometry.
Two lines intersect such that the measure of each obtuse angle formed by the lines is three times the measure of each acute angle formed by the lines. What is the measure, in degrees, of an obtuse angle formed by the lines?
Re: geometry
Hi;
3x + x = 180
4x = 180
x = 45° that is the acute angle.
3x = 3 * 45 = 135° is the obtuse angleHi;
I am getting 92° .Asymptote, that is a program that another forum usesThere are online drawing programs and there is the best, Geogebra
Question: What is the name of the program mentioned in the text that is used on another forum? Answer: Asymptote | 677.169 | 1 |
Multiplying this out, we are left with ½a + ¼a + ¼b - ¼a, which simplifies to ½a + ¼b. Hurrah!
Magnitude
The magnitude of a vector is its length. A unit vector, having a length of 1, will therefore have a magnitude of 1. The magnitude of line OA would be written in pipes (like the modulus) as so:
→ |OA|
To calculate the magnitiude of a line, we can use Pythagoras' Theorem, which works in both 2D and 3D. Given a line AB, with co-ordinates (x, y, z), then |AB| = √(x² + y² + z²). (Omit z if working in two dimensions)
The Scalar Product
The scalar product of a pair of vectors, a and b, is the product of their magnitudes multiplied by the cosine of the angle between them. From this definition, we can write this as a . b = |a| |b|cosΘ. This can be used to find the angle between the two lines, by dividing by |a| |b| and taking the inverse cos, leaving Θ. This can be used to quickly determine if two lines are perpendicular; use a . b = 0, as cos-10 = 90°.
As we write this as a . b, the scalar product is also referred to as the dot product.
Question: If the scalar product of two vectors is zero, what does it mean about the angle between them? Answer: It means the vectors are perpendicular (cos^-1(0) = 90°) | 677.169 | 1 |
Solution: We could do this calculation using the principle of inclusion and exclusion, but what happens if 3 is replaced by 10 in the original question? The calculations would get really ugly, really fast. Instead, we can efficiently calculate the complement probability. The probability that we do not see 6 after rolling 3 times is simply , since each roll is independent and there is a chance that we don't get a 6 on each roll. So the probability that we do see a 6 is . If we replaced 3 with any number in the question, we can easily see how our answer will change.
Test Yourself
1. An integer is chosen at random from 1 to 100 inclusive. What is the probability that it is both a multiple of 2 and a multiple of 5?
2. An integer is chosen at random from 1 to 100 inclusive. What is the probability that it is either a multiple of 2 or a multiple of 5?
3. Eight six-sided dice are rolled. What is the probability that at least two of the dice showed a 1?
4. 10 six-sided dice are rolled. What is the probability of having at least 3 dice showing the same number?
5. (*) A fair coin is repeatedly flipped. What is the probability that 3 consecutive heads will appear before two consecutive tails?
The trigonometric functions are functions of an angle, the most prominent of which are sine, cosine and tangent. These are best understood by considering the line segments from a unit circle.
Given an angle , construct the half ray, , that is an anti-clockwise rotation of from the positive x-axis. The intersection of this ray with the unit circle, labelled as is given by , which gives the definition of . We further introduce that .
From this point , let's drop a perpendicular to the x-axis, intersecting at . Then, is a right angled triangle with . Hence, if we are given any triangle where and , then by similarity with triangle , we get that
If you need a mnemonic to remember these formulas, most english speakers use "Soh Cah Toa". [Personally, I remembered it as "Toa Cah Soh", which means "Pig trotters women" in Cantonese.] In either case, the first letter stands for the trigonometric function, the second letter stands for the numerator, and the third letter stands for the deonominator. We refer to as the Hypotenuse, as the Opposite side and as the adjacent side, and the mnemonic states that
There are certain values of these functions which are useful to remember. They are:
The reason for writing them in this way, is to aid remembering these terms. For example, the numerator for is simply the square root of 0, 1, 2, 3, 4.
Worked Examples
1. What is the graph of ?
Question: What is the probability that an integer chosen at random from 1 to 100 is either a multiple of 2 or a multiple of 5? Answer: 1 - (1/2) * (1/5) = 1 - 0.1 = 0.9
Question: What is the value of sin(0 degrees)? Answer: 0
Question: What is the formula for sine in a right-angled triangle? Answer: Opposite side / Hypotenuse | 677.169 | 1 |
So, what can you do with Kig? Is it just a geometry teaching tool? Kig would be interesting if it were only for teaching geometry, but it can be used for much more. I can easily see how to use Kig to teach algebra, geometry, trigonometry, physics, analysis and calculus.
Let's start with algebra. Figure 1 shows two lines on the Cartesian (X,Y) coordinate plane. Each line is defined by two points, and the coordinates of those points are displayed nearby. The red point in the center is the intersection of the two lines. The equations of the lines also are displayed. By dragging the points around, you can change the lines and explore concepts such as slope, Y-intercept, the solution of an equation and the solution of a system of two equations.
I vaguely remember something from high-school geometry that said "the opposite interior angles formed by two parallel lines crossed by a third, are equivalent." Figure 2 demonstrates this statement for the case where the angles happen to be 90 degrees. Kig makes it easy to construct two parallel lines. Then, you create a third line that crosses the other two. Finally, you tell Kig to label the angle formed by the various lines. Once this is done, you or your student can change the angles, the distance between the parallel lines and so on—the theorem still holds.
I also remember from when I was in school how obtuse mathematical statements tend to be (particularly those statements related to triangles and the size of their sides and angles), but quite often a picture easily much explains them. Kig would allow you to create an interactive demonstration of each the Euclidean geometry theorems.
Measurements of 30-60-90 do not make a very attractive super model, but they do make a great triangle (at least in Euclidean space, but let's not warp things too much here). Figure 3 demonstrates that the sum of all angles in a triangle is 180 degrees. Once this diagram is constructed, students can drag the angles around and explore any triangle they please.
This also would be a great way to demonstrate the various trigonometric ratios, such as sin, cos and tan. In this case, you simply would construct a right angle, and let the student manipulate the lengths of the sides. Kig could be asked to display the angles and lengths of the sides, and the student then could calculate and verify the various ratios.
But, this is where I found what I think to be one of the weaknesses of Kig. Perhaps I just don't know how, but I was unable to create a triangle and explicitly configure two of the angles. I could drag points to the approximate position I wanted, but I was unable to construct an exact 30-60-90 triangle. It seemed like Kig was keeping the lengths of the sides constant, and thus, when I tried to change the angles, they just didn't "fit". I think this is important, so if it can be done, please let me know how.
Question: Which trigonometric ratios can be explored using Kig with a right-angled triangle? Answer: Sin, cos, and tan.
Question: What theorem is demonstrated in Figure 2 using Kig? Answer: The theorem demonstrated in Figure 2 is that the opposite interior angles formed by two parallel lines crossed by a third are equal.
Question: Is it possible to create an exact 30-60-90 triangle using Kig? Answer: It is not explicitly stated in the text that it's possible, but the author expresses difficulty in doing so.
Question: How can Kig be used to teach algebra? Answer: Kig can be used to teach algebra by allowing users to manipulate lines on a Cartesian coordinate plane, defined by two points each, to explore concepts like slope, Y-intercept, and solving equations. | 677.169 | 1 |
angles of a convex polygon having n sides is (n-2)*180.
• Prove that the sum of the measures of the
exterior angles (one at each vertex) of a convex polygon is 360.
• Using modeling rather than memorization to
determine the sum of the measures of the interior angles of a convex
polygon with n sides.
• Solve problems that require combinations of geometric concepts.
• Visualize three dimensions figures in order to
count the number of faces, vertices, edges, and diagonals associated
with these figures.
• Define and differentiate between regular polygons and solids and those that are not regular.
• Use knowledge of nets to determine
characteristics of the unseen sides of a cube given a net of the cube.
• Model nets for solids other than cubes.
• Sketch and use networks to solve problems.
• Recognize congruence and apply the knowledge to solve problems.
• Verify that triangles are congruent using s.a.s., a.s.a., s.s.s
• Verify that a.a.a is not sufficient to prove triangles congruent
• Use construction tools to perform elementary constructions.
• Use constructions to illustrate triangle congruences and similarities
• Use a Mira and paper folding for constructions
• Verify the Pythagorean theorem using construction and scissors
• Recognize the converse of a theorem
• Use the Pythagorean theorem and its converse
• Find surface areas of geometric solids
• Find volumes of geometric solids
• Verify the conversion factor between Centigrade and Fahrenheit
• Perform translations, reflections, and rotations by constructions, using dot paper and tracing paper
• Perform compositions of transformations
• Perform size transformations
• Analyze figures to determine symmetries
• Tessellate a page using a combination of transformations
• Discover properties of altitudes and medians of triangles.
• Prove or verify that constructions actually accomplish the required outcomes
• Discover and list properties of quadrilaterals
• Discover, list, and use properties of similar triangles
• Separate a line segment into n congruent parts by construction and by using lined paper
• Use the Cartesian coordinate system to determine slopes of lines
• Use the factor/label method for measurement conversions
• Use dot paper to find areas
• understand measurable attributes of objects
• identify the units, systems, and processes of measurement
• apply appropriate techniques, tools, and formulas to determine measurements
• Use indirect measurement to solve problems
• Justify area formulas for triangles, parallelograms, and trapezoids
• Find areas of regular polygons
Question: Which of the following is a method to find the area of a figure? A) Using a ruler B) By counting the number of squares C) Using dot paper D) By measuring the perimeter Answer: C) Using dot paper | 677.169 | 1 |
the screenshot its the representation of the reality of a part of the city of buenos aires, the streets are not really a straight grid i guess so i dont know whats going on with voroni but its betraying me
you can use area on closed curves to. Its not the issue. The issue is that voronoi is a perfect radius from each point. those "sites" are not perfect as say a bunch of the same rectangle arrayed and evenly spaced. I dont think voronoi is the answer, maybe im wrong.
Question: What can be used instead of Voronoi, according to the speaker? Answer: The speaker does not suggest an alternative, only expresses doubt about Voronoi | 677.169 | 1 |
Great question, Surajalok - this brings up a crafty GMAT writing style that I've always admired (while watching students struggle with it).
The perimeter of an isosceles right triangle can be calculated using the fact that the ratio of the sides will be x, x, and x*sqrt 2. So, the perimeter will be 2x + x*sqrt 2. We can set that equal to the given perimeter:
2x + x*sqrt 2 = 16sqrt 2 + 16
This one is probably more difficult to solve algebraically than conceptually, so let's just look at the concept. We know that one of the terms (either 16 or 16sqrt 2) will be divided into the two shorter sides, and the other term will represent the longer side (the hypotenuse). So, our options are:
16 represents the two smaller sides; 16sqrt2 represents the long side
OR
16sqrt 2 represents the two smaller sides; 16 represents the long side
If we try the first option, we'd cut 16 in half to create two sides, and our sides would be 8, 8, and 16sqrt 2. This doesn't follow the ratio - the third side, if the first two are both 8, should be 8sqrt 2, so this one doesn't hold.
Trying the second one, we find that the shorter sides are 8sqrt 2 and the long side would be 16. Well, 8sqrt 2 * sqrt 2 fits the ratio, and gives us 8*2 = 16, so the long side fits with the shorter ones.
Therefore, the hypotenuse in this problem has a length of 16.
What I love about this problem (and the many like it) is that the authors of the test know that, because of the ratio x-x-x*sqrt 2, we like to see our hypotenuses end with a square root of 2 on the end. However, if the shorter sides carry a sqrt 2, then the ratio will multiply it out of the hypotenuse, which will then be an integer. It doesn't seem to look right when you're viewing the answer choices, but the math holds up. So...be careful when using right triangle side ratios (45-45-90 and 30-60-90) that you don't preemptively commit to having the radical sign assigned to the side that carries it in the ratio - the GMAT is great at creatively moving the radical to throw you off
Question: What is the given perimeter in the provided problem? Answer: 16sqrt 2 + 16
Question: Which of the following options is correct for dividing the given perimeter into sides of the triangle? A) 16 represents the two smaller sides; 16sqrt2 represents the long side B) 16sqrt 2 represents the two smaller sides; 16 represents the long side Answer: B | 677.169 | 1 |
of points on the line segment between P1 and P2.
For an example, let us consider
the equation in two dimensions Y = a·X + b.
As a final remark, consider the
point Q = ( b/a ·
(a
– 1), a·
b) for a
= 2.0.This becomes
Q = ( b/a , 2 ·
b) or X = b/aand Y = 2 ·
b.For X = b/a, the linear equation
evaluates to
Y = a·(b/a)
+ b b + b = 2 · b, so the point Q = ( b/a
, 2 ·
b) is on the line described by the linear equation.It is just not on the line segment between
the two points.
Convexity
We have defined the line
segment between points P1 and P2 as the set of all points
Q such that Q = (1 – a)· P1 + a· P2, with 0.0 £a£
1.0.Equivalently, we have given the
definition as Q = a1· P1
+ a2·
P2, with a1³
0, a2³
0, and a1
+ a2
= 1.0.In general, we extend this
concept to N points, P1 PN, by defining Q = a1· P1
+ a2· P2
+ + aN· PN,
with the two constraints
a1³ 0, a2³ 0, , aN³ 0, and
a1
+ a2
+ + aN
= 1.0.
The set of all points Q so
formed is called the convex hull of
the point set P1, P2, , PN.There are a number of ways to imagine the
convex hull.Let H be a polygon with
vertices a subset of the points P1, P2, PN.We pick the vertices of the polygon H so as
to make it convex, specifically the line segment between any two of the points
in P1, P2, , PN is included entirely in the
interior of H.Then H is a convex hull
of the point set.
With the informal definition
just given, it is not obvious that the convex hull of a point set is uniquely
defined.Given the formal definition,
the claim that the convex hull is uniquely defined is at least a bit more
plausible.It may even be true.
Dot and Cross Products of
Vectors.
When we discussed the three
elements of an affine vector space, we indicated that there were additional
operators on vectors that were not necessary for the formal definitions.We now discuss the two standard vector
products and later specify convenient methods for their computation.The two products are the dot product and the cross
product.Let U and V be two
vectors.The dot product is denoted U·V and the cross product U
x V.
The properties of the dot and
Question: How is the convex hull of a point set defined for N points? Answer: Q = a1· P1 + a2· P2 +... + aN· PN, with the constraints a1³ 0, a2³ 0,..., aN³ 0, and a1 + a2 +... + aN = 1.0
Question: What is the equation given for a line in two dimensions? Answer: Y = a·X + b | 677.169 | 1 |
This is great work, consider it swiped! Your math all looks right on to me. Although technically, leaving a radical in the denominator is frowned on, I see how it makes the patterns easier to see, and subsequently memorize if desired.
Inspired by your example, I have drawn (on paper) a similar graph of my own. Mine differs in that it is divided only into twelfths with solid lines and eighths in dotted lines, leaving out some of the lesser-used 2-dozenths.
I also labeled each "spoke" with its dozenal point value of tau. In other words, 30 degrees on standard decimalized chart = Tau/12 on yours = .1Tau on mine. 45 degrees = Tau/8 = .16Tau, 90 degrees = Tau/4 = .3Tau, 300 degrees = 5Tau/6 = .A Tau, 330 degrees = 11/12 = .B Tau, etc. This notation really brings home to me the advantages of both tau and dozenal in the unit circle.
As a first attempt, my chart looks like a combination of a clock and a pizza, which doesn't sound appetizing at all, but tastes like dessert to the mind. Looking at it makes everything just about the unit circle "click" for me.
So hey... thanks.
Fashion guru Kate Moss is able to control the temperature of hot tight jeans, sweetheart singer Britney Spears is determinedly sought after low-waist denim. Era in power in all the tight-fitting, low-waist jeans, you need to continue to lose weight, you follow the trend to buy that designer jeans are barely able to wear? This year, in the end people wear pants, or pants to wear it?
Question: What are the two types of lines used in the author's graph? Answer: The author's graph uses solid lines for twelfths and dotted lines for eighths. | 677.169 | 1 |
Question 208377: this is the question: Describe a real-life example of three lines in space that do not intersect each other and no two lines lie in the same plane.
this is what I think will work but I don't know: a book.
Please help!! Click here to see answer by Alan3354(30924)
Question 210389: i have to find the length of d - one side of a triangle, in the simplest radical form. The triangle has a 30 degree angle, a 60 degree angle and a 90 degree angle. side d is the lenght between the 30 and 90 degree angles. Help!!! i have no idea where to start this or how to do this. please help!!! jim_thompson5910(28476) checkley77(12569)
Question 229
Question: What is the length of side 'd' in a 30-60-90 triangle, if the length of the side opposite the 30 degree angle (short leg) is 6 units? Answer: 6√3 units. | 677.169 | 1 |
At that level would the answer not be a bit more pattern related. I.e that the length of the diagonal on 3 is the same as 2 +1 and 5 is the same as 2+3or 4+1. Or that the diagonal will always be the length of the side x the diagonal of the 1cm sided triangle.
You got full marks if you recognised the pattern. The teacher said that ideally the children should have expressed the pattern as an equation (which ds had done). He said he just wanted the kids to think why that pattern might be there. I think this is great but sometimes I think that schools overlook the fact that parents did not have homework in primary school and are therefore often slightly flummoxed by it.
I like the graph paper answer. Get DS to draw the 1cm square (call it S) and cut it in half to make the triangle (call it T). Next get DS to draw the 2cm triangle and break it down into 1cm chunks: you can see that it is one S and two T. The 3cm triangle is three S and three T. The 4cm triangle is six S and four T. etc etc. You can actually see and count the little hypotenuse along the hypotenuse of the bigger triangle.
I love Maths: you can approach it so many different ways but it still comes back to the same answer!
Question: What is the pattern observed in the length of the diagonal of a square as its side length increases? Answer: The diagonal of a square is the same as the sum of the side lengths of the 1cm sided triangle and the square itself. | 677.169 | 1 |
Writing up constructions involves two steps: a construction or "recipe",
where you state precisely all the steps of the construction, and secondly
a proof that the construction does what you claim it does.
Once you have done a construction once, you may refer to it everafter.
So if you have proved that you have a constuction for an angle bisector,
then after that you may use a step such as "Construct the bistector
of angle PQX" and justify it by saying "Page 61, #2."
CONSTRUCTIONS STEPS:
The possiblities for the recipe steps are limited by the construction
axioms. The first two of these types of recipe step are not listed in
the drawing postulates. (Why? Maybe because they were considered too
basic.) The remaining types of step are justified by the Postulates and
by combining previously understood steps.
(1) You may label (give a name to) any point you have drawn, or any
point where two lines or circles intersect.
(2) You can draw an arbitrary point (e.g., not on any existing line
or circle). Be careful that you don't make any assumptions about such
a point. For example, you can't just "draw" the midpoint of a line segment,
though you may be able to do so with some sort of constructions.
(3a) You may draw a line through two points (by Postulate 1).
(3b) You may draw a line through a given point, but otherwise arbitrarily.
Technically you can do this by combining steps of type 2 and 3a as follows.
Call the given point P; choose an arbitrary point Q (type 2); draw the
line PQ (type 3a).
(3c) You may draw an arbitrary line. (Two steps of type 2 and one of
type 3a).
(4a) You may draw a circle with a given center and a given length as its
radius (Postulate 3). Note: for a length to be "given", it must be
the length between two given points on your paper. If someone has
drawn a line segment you may not assume the endpoints are marked, and
you may not draw in the endpoints. In fact if someone draws a segment
on your paper with no points marked, the Greeks would consider that
you have been given a line, not a line segment (since by Postulate 2,
the segment may be extended in a line as far as you want).
(4b) You may draw a circle centered at a given point with an arbitrary
radius (Call the given point P; choose an arbitrary point Q and then
draw the circle centered at P with radius PQ).
(4c) You may draw an arbitrary circle (Choose P arbitrarily and proceed
as in step 4b).
PROOF STEPS:
Some steps of a proof are simple assertions, justified as "by construction".
For example, if you have drawn a circle through A with radius BC and then
Question: What are the first two types of recipe steps not listed in the drawing postulates? Answer: The first two types of recipe steps not listed in the drawing postulates are labeling a point and drawing an arbitrary point.
Question: What is the third type of recipe step and how is it justified? Answer: The third type of recipe step is drawing a line through two points, which is justified by Postulate 1.
Question: What is the purpose of the proof steps in a construction? Answer: The purpose of the proof steps is to demonstrate that the construction actually accomplishes what it was intended to do, as stated in the recipe.
Question: Can you use a previously proven construction in a later step? Answer: Yes, you can refer to a previously proven construction in a later step and justify it by citing the page and number where the proof was first presented. | 677.169 | 1 |
chosen a point D on the circle, the equality of lengths "AD = BC" is
justified by construction. Later, when you use a complicated construction
that you have previously proved, you may justify more complicated things
by construction. For example, if you have in your construction
"Step 5: Construct the bistector of angle PQX, call it QR"
justified as
"Page 61, #2",
then a later step may "angle PQR = angle RQX", justified by construction
(if there is any question as to which piece of the constuction justifies
this, you should say, "by construction, step 5").
Justification steps involving theorems (from the Postulates and Theorems
pages or one that you have proved yourself) should be handled similarly.
For example, perhaps you may say "AB = CD", justified by "CPCTC". Here,
you should make sure that a previous step has proved that two triangles
are congruent, and that AB and CD are indeed corresponding sides of the
two triangles. Thus if you are being careful, you'd say "CPCTC, step 7"
where step 7 says something like "triangle ABX is congruent to triangle CDY".
Self-check question: what will the justification for step 7 probably say?
Answer: It will probably be one of the four congruence theorems such as
SAS, and the equalities of the sides and angles you use ought to be
stated as prior steps. In general, when you use a theorem, the hypotheses
of the theorem should be established in previous steps.
When you use one of the "common notions" you don't need to cite it. For
example, if you are trying to establish the hypotheses of SSS so you can
show that triangles ABC and ABE are congruent, you may write "AB = AB"
without citing the fourth common notion "Things which coincide with
one and other are equal". When you use a Basic Theorem it is probably
best for now to cite it, even though they're pretty obvious.
Question: What should you do if you need to justify a step in your construction using a previously proved theorem? Answer: Cite the theorem and ensure the hypotheses of the theorem are established in previous steps.
Question: What is the purpose of the phrase "by construction" in the given text? Answer: To indicate that a statement is true based on the construction process used in geometry. | 677.169 | 1 |
To draw a circle tangent to 3 equal sized circles (this ones for you Husky )
1. Draw 3 circles (same radius) with centers at any given coordinate points. 2. Set up ' layer ' construction and draw polyline (A) to connect all the centers. 3. Draw a line (B) > using Relative Angle ( LR) set to 90 and at an arbitrary length (say e.g. 100) and 'snap' to the center/middle of one line to produce a line perpendicular to it (90 degree right angle). 4. Repeat for the other line (C), the intersection of these two lines provides the center point (D) for the 4th circle. 5. From any circle draw a line (E) from its center to the intersection, this will be the radius for the 4th circle. Job done.
Question: What is the purpose of line A? Answer: To connect the centers of the three initial circles | 677.169 | 1 |
Using the protractor, I make a line from 'B' that is 56 degrees from the vertical and 2.5 inches long. I mark the end of it 'Z'.
I now draw a line between Y and Z, and extend it all the way to where it crosses my course. I measure the angle at Z - This is the Angle on Bow (AOB). In this case, it is 62.5 degrees.
I then measure the distance between Y and Z. The distance between Y and Z is .625 inches (which in our scale is 625 meters), meaning the target is moving at 6.25 knots.
hope this helps to clear the way.... i have since printed it off and added it to my sh3 file
__________________
' We are here on Earth to fart around. Don't let anybody tell you any different.' Kurt Vonnegut guysIf you have the target true course line plotted, AoB can be completely taken out of the equation (distance too for that matter).
You just have to understand how your SH3 TDC works. The TDC will calculate the AoB for you.
Just take a quick look at this patrol report, and pay special attention to the paragraph where I re-explain the Fast-90 concept to Wolf (my XO).
Question: What is the angle between this line and the vertical? Answer: 56 degrees | 677.169 | 1 |
Triangle Centers Coordinate Geometry: Common Core Performance Task performance task assessment at the conclusion of our unit on triangle properties. Ultimately the unit is about making strategic use of mathematical tools, an idea that is explored through the following content items:
* Perpendicular bisectors and angle bisectors.
* Triangle medians and midsegments.
* Triangle centers: circumcenter, incenter, orthocenter, centroid.
* Geometric constructions: compass & straightedge, paper folding, geometry software, and coordinate geometry.
This particular assessment evaluates several content items and their skills with the coordinate geometry tool. I have included three versions of the assessment, to accommodate varying levels of student ability: A, B, and C, in ascending order of difficulty. Answer keys and student rubrics are included. Aligned with common core standards for high school geometry.
Also included is a cover sheet with common core standards alignment, student objectives, essential questions, and teacher notes.
The complete file (minus answer keys) is available for free preview on the right. Feel free to take a free look before you purchase.
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Be sure that you have an application to open this file type before downloading and/or purchasing.
14182.26Mary Sieczkowski
I have only used Part A (all three pages) but it was a great revisiting of medians, mid segments & perpendicular bisectors. We are getting ready for winter final exams and it helped clarify some information we had covered quickly. The visual, actually drawing the segments really helped a lot of the class.
As a collaborative assessment, you should allow about 30 minutes, but I'd say that's a ball-park estimate. You should consider level of students, comfort with the content, and experience with this type of assessment when blocking time in your schedule. I always find that each class presents different needs. Some need lots of coaxing to get those verifications and justifications out onto the paper. I hope it goes well for you.
Question: According to the text, which of the following is a benefit of using this assessment? A) It helps clarify previously covered information B) It requires no prior knowledge of the topic C) It is only suitable for advanced students Answer: A) It helps clarify previously covered information | 677.169 | 1 |
We are working our way through the Level C Lessons concerning triangles, circles, and fractions (lessons 69-74).
Concerning the dividing of a triagle into fourths, is he supposed to know how to find the midpoint of a number such as 7 inches or 3 1/2 inches or is he just supposed to "eye it"? This is pretty easy to do when the triangle is cut out and can be folded but is more difficult when using a pencil and ruler. Any tips to help my child find the midpoint with difficult measurements such as half of a 3 1/2 inch equalateral triangle...or is this not expected at this point for an 8 yo boy?
Concerning the dividing of equalateral triangles into thirds, is my son expected to know how to divide 3 1/2 inches in half to find the midpoint or is he just supposed to "eye it"? I think I understand that he is to use the 30- 60 triangle point to draw the straight lines to divide the triangle in half 3 times (at each point) but is he supposed to know WHY he uses the 30 degree angle?
Concerning the dividing of the circle into thirds, is he just supposed to "eye" this as well or is there a trick to dividing the circle into thirds correctly? I know that it is supposed to resemble the triangle as divided into thirds but don't know if it is to be more precise.
As far as finding the midpoint of the triangle goes -- no measurement is needed if you use your drawing tools. You can use your T-square and 30/60 triangle to get the midpoint. If you line the triangle or T-square with the point (apex) of the triangle (depending on which way the triangle is pointing), this will give you the midpoint, much the same way that folding the triangle in half gives the midpoint.
I am not sure how much of the "why" he needs to understand about why he uses the 30 degree triangle -- later he will use the 60 degree triangle. We took the try it and see which one works better approach!
Concerning dividing the circle into thirds -- you start by dividing it in half, much like he did with triangle (use the dot in the center of the circle, so that he knows where the line is supposed to go through). Then, using the triangle, he can draw the other two lines, both of which are to end in the center of the circle at the dot (the dot is pre-drawn on the worksheet).
Note on lesson 71, there are two ways to divide the triangle into thirds -- one way gives triangles, the other way quadrilaterals. I missed this at first and had a difficult time figuring out final part of the lesson, until it was pointed out to me -- maybe I was going to fast!
Thank you for the help. I still have a question, though.These lessons are to train for critical thinking/problem solving skills. The child is not expected to already know this, they should be exploring and discovering this.
Question: Are these lessons meant to train the child to already know the methods of division?
Answer: No, these lessons are meant to train the child to explore and discover these methods, not to already know them.
Question: What are the main skills these lessons aim to develop in the child?
Answer: These lessons aim to develop critical thinking and problem-solving skills in the child.
Question: What are the two ways to divide a triangle into thirds mentioned in lesson 71?
Answer: The two ways to divide a triangle into thirds in lesson 71 are: one way gives triangles, and the other way gives quadrilaterals. | 677.169 | 1 |
Problem 116.
Area of Triangles, Excircles.
Level: High School, SAT Prep, College
In the figure below, given a triangle
ABC, construct the excircles with
excenters P and Q. Let be D and E the tangent
points of triangle ABC with its excircles. PE and DQ meet at F,
BE and DQ meet at H, PE and BD meet at G. If S1, S2, S3,
and S4 are the areas of the triangles BPG, EFH, DFG, and
BHQ
respectively, prove that S1 + S2 = S3
+ S4.Post a comment or solution.
Question: What is the level of the problem as indicated in the text? Answer: The level of the problem is High School, SAT Prep, and College. | 677.169 | 1 |
It is evident that there are many lines parallel to the base CD of the triangle, namely BE and LG, as well as innumerable ones in the smaller pentagrams. That means that there will be many 36°-72°-72° triangles besides the large one ACD. The next smaller one is ALG, then FKH, then many smaller ones. Also, each of these various sized 36°-72°-72° triangles is congruent many others in the diagram. For instance, triangles ALG and EAK are congruent.
There are also a series of obtuse 36°-36°-108° isosceles triangles of varying sizes.
All these parallel lines and similar triangles yield numerous relationships among the various diagonals and sides of the pentagons. Some of these relationships are additive equations:
d1 = s1 + d2 s1 = d2 + s2 d2 = s2 + d3 s2 = d3 + s3
and so forth.
Other relationships are based on the property of 36°-72°-72° triangles used in their
construction in IV.10, namely that the square of the base of such a triangle equals the product of a side and the difference between the side and the base. In terms of the diagonals and sides of the pentagons, this gives the equations:
d1d2 = s12 s1s2 = d22 d2d3 = s22 s2s3 = d32
and so forth.
After ratios are proportions are developed in
Book V and Book VI, we can add the following
continued proportion to the list of relationships:
d1:s1 =
s1:d2 =
d2:s2 =
s2:d3 =
...
See the Guide to proposition X.2 which shows that diagonal and side of a regular pentagon are incommensurable. In more modern terms we would say that their ratio, which is called the "golden ratio," is an irrational number.
Use of this proposition
This construction is used in the next proposition to circumscribe a regular pentagon around a circle and later in IV.16 to construct a regular 15-gon. It is also used in XIII.16 for the construction of a regular icosahedron (a 20-sided polyhedron each of whose faces is an equilateral triangle). Surprizingly, it is not used in XIII.17 for construct a regular dodecahedron (a 12-sided polyhedron each of whose faces is a regular pentagon); the regular pentagons needed for it are constructed in space directly without the help of this proposition.
Question: Which regular polyhedra are constructed using this proposition? (Choose all that apply) Answer: Regular pentagon, regular 15-gon, regular icosahedron
Question: What is the angle measure of the obtuse triangles mentioned? Answer: 36°-36°-108°
Question: What is one of the additive equations relating the diagonals (d) and sides (s) of the pentagons? Answer: d1 = s1 + d2 | 677.169 | 1 |
Space Probes: Scientists are predicting a comet will collide with an asteroid in 30 days. Three different probes can be sent…(math skills used: equation of a line / standards 1,4,5)
Batter Up: Troy is a very good batter. He averages a hit every one out of three times he comes to bat. If Troy batted twice…(math skills used: probability / standards 5)
WiFi Access: The owner of a campground would like to install WiFi (wireless internet) access for his customers. He currently…(math skills used: distance formula, midpoint formula / standards 2,3)
Golden Corrals: Robert has thirty corral panels that measure 10 feet each. He originally set up the corral in a rectangular shape…(math skills used: Pythagorean thm, perimeter, area / standards 1,2,3)
Experimental Data: The data points in the table below came from an experiment that involved hydrogen (H) and carbon (C). All measurements…(math skills used: equation of a line, experimental data, best fit lines / standards 4,5)
Question: Which mathematical theorem is used in the "Golden Corrals" scenario? Answer: Pythagorean theorem | 677.169 | 1 |
Apolyhedron is a three-dimensional solid
figure in which each side is a flat surface. These flat surfaces
are polygons and are joined at their edges. The word
polyhedron is derived from the Greek
poly (meaning many) and the
Indo-European hedron (meaning seat
or face).
A polyhedron has no curved surfaces.
The common polyhedron are pyramids and
prisms.
pyramid
prism
A polyhedron is called
regular if the faces are congruent, regular polygons and the
same number of faces meet at each vertex. There are a total of
five such convex regular polyhedra called the
Platonic solids.
tetrahedron
octahedron
icosahedron
hexahedron
dodecahedron
Euler's Polyhedron Theorem: Euler discovered that
the number of faces (flat surfaces) plus the number of vertices (corner
points) of a polyhedron equals the number of edges of the polyhedron
plus 2.
F + V = E + 2
Non-Polyhedra The following solids are not polyhedra since a part or all
of the figure is curved.
Cylinder
Cone
Sphere
Torus
A torus is a "tube
shape". Examples include an inner tube, a doughnut, a tire and a
bagel. Small r is the radius of the tube and capital R
is the distance from the centre of the torus to the center of the tube.
Question: Is a polyhedron a two-dimensional or three-dimensional figure? Answer: Three-dimensional
Question: What are the two common types of polyhedra mentioned in the text? Answer: Pyramids and prisms | 677.169 | 1 |
the angles in the polar coordinates of a point in the 3D world. Eo
stands for equatorial and it is the angle on the xw,yw
plane. Vo stands for vertical and is the angle between
the line from the origin to the point's (xw,yw)
coordinates and the line from the origin to the point's (xw,yw,zw)
coordinates, in other words, the angle on the plane perpendicular
to the (xw,yw) plane.
The sphere is centered at the origin and the xw, yw,
and zw axes are defined in the image below where xw
is blue, yw is yellow, and zw is green. The
black line is the left and right edges of the image stretched
around the sphere. The red disk is the plane made up of only the
xw and yw axes, it also represents the
"equator" of the sphere.
Figure 4: The
axes of the imaginary 3D world
We can get the polar coordinates of any point in the 3D world
(see figure 5) by first measuring the angle Eo the
point's (xw,yw) coordinates make on the (xw,yw)
plane (the red disk) and getting the distance of the line L (the
pink line) it makes to the origin. We can then use the length of
L to find the angle Vo between line L and line R,
which is the line that goes directly to the point's (xw,yw,zw)
coordinates. For instance we can take a point on the sphere which
has the coordinates (x1w,y1w,z1w)
and measure its polar coordinates on two planes. These lines are
shown in the following figure. The red line (line R) goes
directly to the point and has the length of the radius of the
sphere, the pink line (line L) goes to the point's x1w,y1w
coordinates only.
Figure 4: The
axes of the imaginary 3D world
We can measure the angles to this point by spliting the
problem into the two planes shown below. The colors are the same.
Now that we have these equations, we can convert from (xw,yw,zw)
coordinates to angles or from angles to (xw,yw,zw)
coordinates. We can also find the distance from the origin to any
point.
Pixels on the sphere
Because of the way the renderer wraps the image around the
sphere, the angle Vo to every pixel in a horizontal
row of pixels is the same and the angle Eo to every
pixel in a vertical row is the same. Also, there is always the
same number of pixels in a row and the same number of pixels in a
column. We can then find the polar coordinates for any pixel in
the sphere with the following equations, Ih is the
height of the image in pixels.
Vo = (180o / Ih) * (Ih /
2 - yi)
Question: What is the angle Vo between? Answer: The angle Vo is between the line from the origin to the point's (xw,yw) coordinates and the line from the origin to the point's (xw,yw,zw) coordinates.
Question: How can we find the polar coordinates of any point in the 3D world? Answer: We can find the polar coordinates of any point in the 3D world by first measuring the angle Eo the point's (xw,yw) coordinates make on the (xw,yw) plane and then using the length of line L to find the angle Vo between line L and line R.
Question: On which plane is the angle Eo measured? Answer: The angle Eo is measured on the xw,yw plane. | 677.169 | 1 |
1) Always start on the most complicated side
2) Change everything to sine and cosine. These are the trig functions all students are most familiar with, it will be easy to see patterns with them
3) Note that you may have to work on both sides. Working on one side and trying to get the other side does not always work. If you find you can't get the other side, stop and work on the other side to get where you left off on the first side.
Question: Which trigonometric functions are suggested to change everything to? Answer: Sine and cosine | 677.169 | 1 |
Solution of triangles
Solution of triangles (Latin: solutio triangulorum) is the historical term for the solving of the main trigonometric problem: to find the characteristics of the triangle (three angles, the lengths of the three sides etc) when some (but not all) of this characteristics are given. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc.
In a general form triangle, there are 6 main characteristics (see picture): 3 linear (side lengths ) and 3 angular (). The classical plane trigonometry problem is to specify 3 of the 6 characteristics and to determine the three others. Obviously, if we know only 2 or 3 angles, solution is undefined, because any triangle similar to a solution is the solution also, so we assume that at least one of the known values is linear.
Thus a triangle can be solved when given the any of the following information:[1][2]
To find an unknown angle, law of cosines is safer than law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if , the angle can be equal either or . Using the law of cosines avoids this problem: within the interval from to the cosine value determines its angle unambiguously.
We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will be the solution also. For example, three side lengths uniquely define either a triangle or its reflection.
The procedure for solving an AAS triangle is same as that of an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines.
General form spherical triangle is fully determined by three of its six characteristics (3 sides and 3 angles). Note that the sides of a spherical triangle are usually measured rather by angular units than by linear, according to corresponding central angles.
The above algorithms become much simpler if one of the angles of a triangle (for example, the angle ) is the right angle. Such spherical triangle is fully defined by its two elements, the other three can be calculated using Napier's Pentagon or the following relations.
Suppose you want to measure the distance from shore to remote ship. You must mark on the shore two points with known distance between them (base line). Let are the angles between base line and the direction to ship.
From the formulas above (ASA case) one can define the length of the triangle height:
This method is used in cabotage. The angles are defined by observations familiar landmarks from the ship.
How to measure a mountain height
Another example: you want to measure the height of a mountain or a high building. The angles from two ground points to the top are specified. Let be the distance between tis points. From the same ASA case formulas we obtain:
Question: Why is the law of cosines safer than the law of sines? Answer: The value of sine for an angle does not uniquely determine that angle, while the value of cosine within the interval from 0 to π does.
Question: What are the three linear characteristics of a triangle? Answer: Side lengths
Question: What is the procedure for solving an AAS (Angle-Angle-Side) triangle? Answer: First, find the third angle using the angle sum property, then find the other two sides using the law of sines.
Question: In a general form triangle, how many main characteristics are there? Answer: 6 | 677.169 | 1 |
I know that h = 4+9 = 13. I was trying to use the Triangle Angle Bisector Theorem but I don't seem to have enough information. Click here to see answer by solver91311(16897)
Question 186535: THis is greek to me, where would i start with something such as this!:-}
Question 186566: Regarding Question 186543, I got your answer and I appreciate it very much, but if its not too much trouble could you please show me how you arrived at the number 12 for the AC, I hope I can follow it from there. Click here to see answer by jojo14344(1512)
Question 188644: This is a triganometry problem that deals with SOHCAHTOA
To find the height of a pole, a surveyor moves 100 ft. away from the base of the pole and then, with a transit 5 ft. tall, measures the angle of elevation to the top of the pole to be 29 degrees. What is the height of the pole? Round answer to the nearest foot.
Thanks for your help!!!! Click here to see answer by Mathtut(3670)
Question: What is the height of the transit used by the surveyor? Answer: 5 ft | 677.169 | 1 |
Related Products
What is the figure that will result from the cutting of a right circular cone by a plane parallel with the cone's axis of symmetry?
(A) a loxodrome (B) an ellipse (C) a catenary (D) a parabola
Answer is (D)
Solution: The figure that results is a parabola, also known as a conic section. Vertical curves are parabolic curves. This is problem 71(1-63 .
CORRECTED SOLUTION FOR PROBLEM 71(1-63)
POB extends our thanks to Troy J. Groth, PE, of Sundquist Engineering, PC, in Iowa, who spotted an error in this week's solution.
The following is the correct answer along with Mr. Van Sickle's explanation:
The correct answer is: a hyperbola
"A parabola is - open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone."
In other words, if the plane that cuts the cone is parallel to the axis of symmetry of the cone (as stated in the question) a hyperbola is created.
Specifically a parabola is the curve for which the distance from a fixed line (the directrix) is equal to its distance from a fixed point (the focus) whereas a hyperbola is the curve for which the difference between the distances to the foci remains constant. Or stated another way the arms of a parabola eventually become parallel to each other, while the arms of a hyperbola always make an angle relative to each other
Question: What is the type of curve that a parabola eventually becomes? Answer: Parallel to each other | 677.169 | 1 |
Question 252762: Here is my question:
My triangle points are A(2,5), B(12,-1) and C(-6,8).
What is the slope of the perpendicular bisector of AB? What is the slope of CL if CL is the altitude from point C?
I have been working on this for a while and appreciate any help with explanation.
Thanks.
Question 250431: A landscaper wants to put a cement walk with uniform width around a garden that measures 20x40 feet, She has enough cement to cover 660 ft squared. How wide should it be to us all the cement? Click here to see answer by drk(1908 rfer(12644 Alan3354(30924)
Question 257998: An architect is designing a museum entranceway in the shape of a parabolic arch represented by the equation y=-x + 20x, where zero is greater than or equal to x which is less or equal to 20 and all dimensions are expressed in feet. on the accompanying set of axes, sketch a graph of the arch and determine its maximum height, in feet. Theo(3458 260181: tell whether each statement is true or false:
if two lines intersect, then the adjacent angles formed are congruent.
my question is: is the statement true or false and what are adjacent angles?? can you explain to me how you got your answer please [email protected](15622)
Question 261257: Find the value of x. Leave your answer in simplest radical form.
Piture is a right trianggle and base is an x which is the bottom and then on the left side of the right triangle is 8 and the right side of the right trangle is 14. Click here to see answer by stanbon(57203)
Question: What is the slope of the perpendicular bisector of line segment AB, given points A(2,5) and B(12,-1)?
Answer: The slope of the perpendicular bisector of AB is -0.2. | 677.169 | 1 |
Quilting Tips: Setting Triangles
When you position quilt blocks on point (also called a diagonal set), you need to fill in the outer edges of the quilt with triangle-shaped cuts of fabric. Triangles along the side of the quilt are called side setting triangles. Corner setting triangles fit the corners of the quilt.
It's much simpler and quicker (avoids the math) to measure the on-point block on the diagonal from point to point. Add 1/2 inch to that measurement and that is the size of the block for your setting triangles.
Forgot to say that the four corner triangles are cut from a rectangular block that is the measurement of the diagonal of the on-point block long by 1/2 that measurement wide. Cut that in half and then cut those two blocks on the diagonal.
There's a typo in the chart: the corner-triangle formula at the bottom of the chart says to divide by 1.141. It should be 1.414 (the square root of two). A spot-check of the chart itself shows that it uses the correct formula.
Thanks for the chart, and thanks to calicoedcat for the alternate method.
by chance can anyone tell me how to set these seams up correctly so I do not get puckers on the quilt top at the point where you stitch the triangle tip? And would the same technique be used when sewing the corners? Any assistance would soooooo be greatly appreciated.
Question: What is the formula to calculate the size of the block for setting triangles? Answer: Add 1/2 inch to the measurement of the on-point block measured diagonally
Question: What are the triangles along the side of the quilt called? Answer: Side setting triangles | 677.169 | 1 |
he trigonometric functions are among the most fundamental in mathematics. They were initially developed to aid in the measurement of triangles and their angles, and they are useful in such practical fields as surveying and navigation. However, their importance to pure and applied mathematics extends far beyond these uses, because they can be used to describe any natural phenomenon that is periodic, and in higher mathematics they are fundamental tools for understanding many abstract spaces.
RADIANS
The primary use of trigonometric functions is in the measurement of angles. Although the ancient Babylonian degree unit of angle measure is still in wide use, in mathematics we prefer to use the radian measure. Given a circle centered at the origin in the Cartesian plane, imagine taking a radius and laying it along the outside circle, beginning at the x axis and going counterclockwise.
This marks out an angle of one radian. Because the circumference of a circle is twice the radius times p, a full circle corresponds to an angle of 2p radians. Thus we get the following correspondences between degree measure and radian measure:
By remembering the correspondence between radians and degrees indicated by the formula above, one may always convert radians to degrees and vice versa with ease by plugging the known quantity into the equation and solving for the unknown quantity. With practice, using radian measure becomes as natural as using degrees, and the use of radians greatly simplifies our work with the trigonometric functions.
TRIGONOMETRIC FUNCTIONS
The trigonometric functions are most easily understood in the context of a circle in the Cartesian plane, in which angles are always measured from the x axis: positive angles are measured in an anti-clockwise direction, and negative angles are measured in a clockwise direction. In this context, the trig functions are defined as follows:
Note that these definitions induce the following relationships:
TRIGONOMETRIC IDENTITIES
The so-called trigonometric identities are actually quite easy to derive algebraically from the definitions given above, and every student of mathematics should derive them all at least once. Thereafter, it will no longer be necessary to memorize them, for you will be able to derive them when they are needed.
Question: If an angle is measured in degrees and you want to convert it to radians, which formula would you use? Answer: (angle in degrees) * (π / 180)
Question: What is the definition of the sine function in a circle context? Answer: The y-coordinate of the point on the circle corresponding to the angle
Question: In the context of a circle in the Cartesian plane, which direction is positive for angle measurement? Answer: Anti-clockwise | 677.169 | 1 |
Finding circle center from two points and an arc length
Finding circle center from two points and an arc length
I'm trying to find the equation for a circle given two points in x, y and the starting angle, arc length, and two points along the circle. I need to find the equation because I need to translate a sprite along the curved path from one point to another.
Now, I can clearly determine the length of all sides as well as the angles. thats not problem. I can use that to make a system of equations and solve them together. This works, but the problem is that I need to be able to do this programmatically, and attempting to solve systems of equations in C++ is proving to be pretty difficult. I was hoping there was a more straight forward/easy way to find the center when two points are know and all the angles/lengths are known.
In thi case the positive x-axis (angle 0) (originating from the center) and the radius from C to A form the starting angle. Now that I think about it, the starting angle doesnt need to be given, it can be derived by which point A or B forms the lesser starting angle.
No, C is the unknown and it can be anywhere. Its directly dependent on A and B. I know point A, point B and the angle of the arc that needs to be drawn between them. I assumed I'd need to provide a starting angle for the arc but now I don't think that's necessary.
In the drawing, C is the center of the circle that describes the arc between A and B. If it's possible, i want to be able to find C (the center of the circle) without solving a system of linear equations as that is proving to be difficult when solving. programmatically.
I'm sorry, i guess I'm not making this clear enough heh. Thansk for the help so far though :)
A
|\
|a\
|--\
|---\
----c C
|---/
|--/
|b/
|/
B
I was hoping there was a more straight forward/easy way to find the center when two points are know and all the angles/lengths are known.
Thanks in advance for any help
Let d be the distance from A to B. If you know A and B you can find its mid-point M. Lets say the vector AB is <s,t>. Then a vector perpendicular to it is <-t,s>. Divide that vector by its length and call the resulting vector V. If you know the radius r you can calculate the height h of your triangle with the Pythagorean theorem using hypotenuse r and leg d/2.
Then if O is the origin, the coordinates of the center are OA+ (1/2)ABą hV, with the sign chosen depending on which side of AB the center is on.
Question: What is the suggested method to find the center of the circle without solving a system of linear equations? Answer: Calculate the midpoint (M) of AB, find a vector perpendicular to AB, divide it by its length to get vector V, and then use the Pythagorean theorem to find the height (h) of the triangle. The center (C) is then OA + (1/2)AB + hV.
Question: What is the main task the user is trying to accomplish? Answer: The user is trying to find the equation of a circle given two points and an arc length to translate a sprite along a curved path.
Question: What is the unknown variable the user wants to find? Answer: The user wants to find the center of the circle (C). | 677.169 | 1 |
Pythagorean Theorem Practice one double sided worksheet with practice Pythagorean Theorem problems.
There are 8 problems:
3 pictures of right triangles with a missing side
2 coordinate systems to find distance between two points
(shows that the distance formula is just Pythagorean Theorem)
3 odd shapes where one must use a right triangle to find the length of a missing side.
The free preview file IS the whole file --- just with "preview" written all over it!.
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
138.26
Question: What is the relationship between the distance formula in coordinate systems and the Pythagorean Theorem? Answer: The distance formula is just the Pythagorean Theorem applied to the coordinates. | 677.169 | 1 |
As an example, if your bottom right hand corner of the paper is one hundred percent, then the bottom left hand corner would be zero percent. The lines which lay horizontally indicate the different amount of your third component. If the top of the triangle is one hundred percent then the bottom would be zero percent. When you are plotting data with points onto your paper you will only need to know your data for component one and three because with this information you will then receive your information for your second component.
An example of using the graph paper is easily shown when you have collected the data in which you need to be plotted to the paper and shown:
How to Use Triangle Graph Paper
If you have a combination of:
1= 25% 2= 15% 3= 60%
You would first place the 25% on the graph upon the base until you reached the 60% then you would following the upward diagonal line on the graph which you give you the resulting answer of 15%.
This triangle graph paper is exactly what you will need when you are intending to plot any of you equilibrium data on paper. If you were using liquid to liquid data you could plot all of your phases known. You can also use the equilateral triangle graph spreadsheet for configuring and analyzing distillation system data.
Using Triangle Graph Paper for Tessellations
Another common use for calculations with triangle graph paper involve configurations called tessellations. Tessellations are used most often to allow a triangle to "tessellate", or make symmetric, a plane.
Tessellations can be found by making two sets of parallel lines at right angles against one another. Tessellations can also be found by conjuring up a few equilateral triangles, with three sets of parallel lines at 60 degrees to one another.
The features of 3 the Types of EMI calculator available in Excel are comprised of some ordinary features for calculation of EMI on loans, computation of statements of monthly interest payable, statement of monthly loan payment, and statement of unpaid loan amount. The calculators also allow for features such as wanted EMI, rounded EMI on loan amounts, and multipurpose activities for setting up loan repayment, loan increases over time and rates according to floating or fixed interest agreements.
Why Download a Personal Loan EMI Calculator?
If you are looking for the best solutions in the 3 Types of EMI calculators offered in Excel, look no further. There are now eleven features or purposes programmed into the multipurpose EMI calculators in Excel. The three main types are now refined in these various functions. Uses for the different EMI calculator types include: home loan eligibility, floating loan rates, fixed rates, vehicle EMI, accelerated repayment of loan, lump sum repayment scheduling and even reverse EMI computation. Everything you need to manage your outstanding loan accounts is available in the Excel EMI calculator options.
Question: What is the percentage represented by the bottom left corner of the triangle graph paper? Answer: Zero percent.
Question: How many sets of parallel lines are used to create tessellations with equilateral triangles? Answer: Three sets of parallel lines. | 677.169 | 1 |
Definition of a Circle
You've known all your life what a circle looks like. You probably know how to find the area and the circumference of a circle, given its radius. But what is the exact mathematical definition of a circle? Before you read the answer, you may want to think about the question for a minute. Try to think of a precise, specific definition of exactly what a circle is.
Below is the definition mathematicians use.
Definition of a Circle
The set of all points in a plane that are the same distance from a given point forms a circle. The point is known as the center of the circle, and the distance is known as the radius.
Mathematicians often seem to be deliberately obscuring things by creating complicated definitions for things you already understood anyway. But if you try to find a simpler definition of exactly what a circle is, you will be surprised at how difficult it is. Most people start with something like "a shape that is round all the way around." That does describe a circle, but it also describes many other shapes, such as this pretzel:
So you start adding caveats like "it can't cross itself" and "it can't have any loose ends." And then somebody draws an egg shape that fits all your criteria, and yet is still not a circle:
So you try to modify your definition further to exclude that... and by that time, the mathematician's definition is starting to look beautifully simple.
But does that original definition actually produce a circle? The following experiment is one of the best ways to convince yourself that it does.
Experiment: Drawing the Perfect Circle
Lay a piece of cardboard on the floor.
Thumbtack one end of a string to the cardboard.
Tie the other end of the string to your pen.
Pull the string as tight as you can, and then put the pen on the cardboard.
Pull the pen all the way around the thumbtack, keeping the string taut at all times.
The pen will touch every point on the cardboard that is exactly one string-length away from the thumbtack. And the resulting shape will be a circle. The cardboard is the plane in our definition, the thumbtack is the center, and the string length is the radius.
The purpose of this experiment is to convince yourself that if you take all the points in a plane that are a given distance from a given point, the result is a circle. We'll come back to this definition shortly, to clarify it and to show how it connects to the mathematical formula for a circle.
The Mathematical Formula for a Circle
You already know the formula for a line: y=mx+by=mx+b. You know that
mm is the slope, and
bb
is the y-intercept. Knowing all this, you can easily answer questions such as: "Draw the graph of y=2x–3y=2x–3"
or "Find the equation of a line that contains the points (3,5) and (4,4)." If you are given the equation 3x+2y=63x+2y=6,
Question: What is the distance from the center of a circle to any point on the circle called? Answer: The distance from the center of a circle to any point on the circle is called the radius.
Question: If you are given the equation 3x+2y=6, what can you do with it? Answer: If you are given the equation 3x+2y=6, you can find the equation of a line that contains the points (3,5) and (4,4).
Question: What is the slope of a line represented by the equation y=2x-3? Answer: The slope of the line is 2. | 677.169 | 1 |
A2. Find all positive integers which divide 1890·1930·1970 and are not divisible by 45.
A3. The function f(x, y) is defined for all real numbers x, y. It satisfies f(x,0) = ax (where a is a non-zero constant) and if (c, d) and (h, k) are distinct points such that f(c, d) = f(h, k), then f(x, y) is constant on the line through (c, d) and (h, k). Show that for any real b, the set of points such that f(x, y) = b is a straight line and that all such lines are parallel. Show that f(x, y) = ax + by, for some constant b.
B1. AB and CD are perpendicular diameters of a circle. L is the tangent to the circle at A. M is a variable point on the minor arc AC. The ray BM, DM meet the line L at P and Q respectively. Show that AP·AQ = AB·PQ. Show how to construct the point M which gives BQ parallel to DP. If the lines OP and BQ meet at N find the locus of N. The lines BP and BQ meet the tangent at D at P' and Q' respectively. Find the relation between P' and Q'. The lines DP and DQ meet the line BC at P" and Q" respectively. Find the relation between P" and Q".
B2. A plane p passes through a vertex of a cube so that the three edges at the vertex make equal angles with p. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto p. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane p.
1. A graph G has n + k points. A is a subset of n points and B is the subset of the other k points. Each point of A is joined to at least k - m points of B where nm < k. Show that there is a point in B which is joined every point in A.
May 27, 2010
1. At time t = 0, a lion L is standing at point O and a horse H is at point A running with speed v perpendicular to OA. The speed and direction of the horse does not change. The lion's strategy is to run with constant speed u at an angle 0 < φ < π/2 to the line LH. What is the condition on u and v for this strategy to result in the lion catching the horse? If the lion does not catch the horse, how close does he get? What is the choice of φ required to minimise this distance?
Question: What is the choice of φ that minimizes the distance the lion gets to the horse? Answer: The choice of φ that minimizes the distance is φ = arctan(v/u), where u is the lion's speed and v is the horse's speed.
Question: What is the condition for the lion to catch the horse? Answer: The lion will catch the horse if and only if u * sin(φ) >= v, where u is the lion's speed, v is the horse's speed, and φ is the angle between the lion's direction and the line LH.
Question: How many planes are there through three vertices of the cube? Answer: There are 20 planes through three vertices of the cube.
Question: What is the product of the numbers 1890, 1930, and 1970? Answer: 1890 1930 1970 = 7,036,910 | 677.169 | 1 |
2. AB and CD are two fixed parallel chords of the circle S. M is a variable point on the circle. Q is the intersection of the lines MD and AB. X is the circumcenter of the triangle MCQ. Find the locus of X. What happens to X as M tends to (1) D, (2) C? Find a point E outside the plane of S such that the circumcenter of the tetrahedron MCQE has the same locus as X.
It is clear from the graph that there are no roots for k < -1, and one root for k = -1 (namely x = -1). Then for k > -1 there are two roots except for a small interval [1, 1+h]. At k = 1, there are 3 roots (x = -2, 0, 1). The upper bound is at the local maximum between 0 and 1. For such x, y = x + 1 - x2 = 5/4 - (x - 1/2)2, so the local maximum is at 5/4. Thus there are 3 roots at k = 5/4 and 4 roots for k ∈ (1, 5/4).
3. Answer: circle diameter AB, where OB is the normal to p
Let B be the foot of the perpendicular from O to p. We claim that the locus is the circle diameter AB. Any line in p through A meets this circle at one other point K (except for the tangent to the circle at A, but in that case A is obviously the foot of the perpendicular from O to the line). Now BK is perpendicular to AK, so OK is also perpendicular to AK, and hence K must be the foot of the perpendicular from O to the line.
1. A conference has 47 people attending. One woman knows 16 of the men who are attending, another knows 17, and so on up to the last woman who knows all the men who are attending. Find the number of men and women attending the conference.
2. For what values of m does the equation x2 + (2m + 6)x + 4m + 12 = 0 has two real roots, both of them greater than -2. 3. Solve the equation sin3x cos 3x + cos3x sin 3x = 3/8. 4. The tetrahedron SABC has the faces SBC and ABC perpendicular. The three angles at S are all 60o and SB = SC = 1. Find its volume. 5. The triangle ABC has perimeter p. Find the side length AB and the area S in terms of ∠A, ∠B and p. In particular, find S if p = 23.6, A = 52.7 deg, B = 46 4/15 deg.
Question: What is the locus of point X as M moves along the circle S?
Answer: The locus of X is a circle with diameter AB, where OB is the normal to the plane of the circle.
Question: For what value of m does the equation x² + (2m + 6)x + 4m + 12 = 0 have two real roots, both greater than -2?
Answer: The discriminant of the quadratic equation is (2m + 6)² - 4(4m + 12). For two real roots, the discriminant must be positive, i.e., (2m + 6)² - 4(4m + 12) > 0. Solving this inequality gives m > 0.
Question: How many women are there at the conference?
Answer: There are 47 - 1128 = -1081 women, which is not possible. This indicates there might be an error in the problem statement or the interpretation of the sequence. | 677.169 | 1 |
Let's take (A),
X > Y means nothing, for e.g. if X = 135 and y = 45, then the line segments are equal. The length will contract and expand from one side of the 90 deg to the other. So if we draw a perpendicular line (i.e. assume x = 90, then the lengths are same for each degree either side). Not useful.
Take (B), as in case (A), X+Y>90 means nothing, as there are many possible variations with either PQ=SR or greater or lesser.
Question: In a right-angled triangle, which angle is always 90 degrees? Answer: The angle that is being referred to as 'X' in the text. | 677.169 | 1 |
Trigonometry
The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.
Trigonometry (from Greektrigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.[4] They and their successors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
Question: What is the name given to the reciprocal of the tangent function? Answer: Cotangent (cot) | 677.169 | 1 |
Plotting a Cycloid
This animation shows how a point on a moving circle generates a cycloid.
(See notes below.)
The curve traced out by a point fixed to a circle as the circle rolls without
slipping along a straight line is called a cycloid. The cycloid frequently appears
in elementary calculus textbooks as an example of how to determine the parametric
equations for a curve given a geometric definition for the curve. See
also the
epicycloid
and the
hypocycloid.
(11/17/07)
Question: What is the key difference between a cycloid and an epicycloid? Answer: While a cycloid is generated by a circle rolling along a straight line, an epicycloid is generated by a circle rolling along the outside of another circle. | 677.169 | 1 |
math From the given circle equation you would have to know that its centre is (0,0) and its radius is √40 Notice that both A and B satisfy the equation of the circle, so A and B lie on the circle and AB is indeed a chord. In a) you are probably expected to illustrate the ...
Thursday, October 9, 2008 at 5:11pm by Reiny
Math (Pre Cal) Hi, I am studying for a precalculus quiz and I do not understand this hw problem: "The tangent line to a circle may be defined as the point that intersects a circle in a single point... If the equation of the circle is x^2+y^2=r^2 and the equation of the tangent line is y...
Saturday, August 28, 2010 at 11:57pm by Dealie
MATH grade 10 so basically you have a circle, right? and the equation of a circle is x^2+y^2 = r^2 so in this case your r, or radius is root of 64, or 8 if the radius of the circle is 8, use the formula for circumference of a circle (c=2pi(r)) to find the length of fencing so... c=2pi(8) so...
Tuesday, October 6, 2009 at 9:25pm by brandon
8th grade math circle A has a raduis that is twice the length og the raduis of circle B. Which is an accurate statement about the relationship of these areas of circle A and B and y? a.the area of circle A is 4 times the area of circle B b. the area of circle A is twice the area of circle B...
Thursday, December 16, 2010 at 10:18pm by Court
Math This question makes reference to an orthonormal basis (i,j), and an origin O. 1. Consider the triangle ABC, with vertices A(0,-2), B(9,1), C(-1,11). Find: a) a cartesian equation of the altitude from C; b) a cartesian equation of the circle which has BC as a diameter; c) a ...
Wednesday, February 27, 2008 at 11:08am by Anonymous
Math Answer check please: Consider the circle (x-3)^2 + (y+1)^2 = 169. The point (15,4) is on the circle. Find an equation for the line that is tangent to the circle at this point. I got y= 5/12x-9/4.
Sunday, August 26, 2012 at 2:44pm by KC
equation for circle find the equation of a circle. the endpoints of the diameter of a circle are (3,-2) and (7,-6).
Sunday, February 5, 2012 at 6:44am by Declan
Question: What is the equation of a tangent line to a circle with equation x^2 + y^2 = r^2? Answer: It is not given in the text, but it's mentioned that it intersects the circle at a single point.
Question: In the context of the first text, what does 'a' and 'b' represent? Answer: 'a' and 'b' represent points on the circle.
Question: If the radius of a circle is 8, what is the circumference of the circle? Answer: 2 π 8 = 16π
Question: What is the center of the circle given by the equation x^2 + y^2 = 40? Answer: (0, 0) | 677.169 | 1 |
as follows: (1) draw a radius of the disk, passing
through p, (2) draw a rectangle with a corner at
the point where this radius meets the boundary of D, and the
opposite corner located on the same radius, just large enough so
that p is in the interior of the rectangle.
Question: What is the location of the second corner of the rectangle? Answer: On the same radius as the first corner, which is at the point where the radius meets the boundary of the disk. | 677.169 | 1 |
-Angles that are supplementary to the same angle or to congruent angles are congruent
-Angles that are complementary to the same angle or to congruent angles are congruent
-All right angles are congruent
-Vertical angles are congruent
-Perpendicualr lines intersect to form 4 right angles
-a + b = c . (angle addition theorem.
A transversal is a line that crosses two parallel lines.
Line m is parallel to n, so p is the transversal. All of the new postulates for transversals involve the angles formed. The angles formed by the transversal are named for their relation to the transversal.
Question: Are all right angles congruent? Answer: Yes | 677.169 | 1 |
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