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Precalculus
the segment perpendicular to the transverse axis of a hyperbola through its center.
4.
the midpoint on a point of a line segment joining the foci.
5.
the axis of symmetry of an ellipse which does not contain the foci.
6.
the study of coordinate geometry form an algebraic perspective.
8.
one of the two segments into which the center of an ellipse divides the major axis.
12.
half of a circle.
15.
a given line.
16.
a line that a graph approaches but never intersects.
17.
a set of points and only those points that satisfy a given set of conditions.
19.
the line segment that has as its endpoints the vertices of a hyperbola.
21.
a special case of the equilateral hyperbola, where the coordinate axes are the asymptotes. the general equation of a rectangular hyperbola is xy = c, where c is a nonzero constant.
23.
a curve determined by the intersection of a plane with a double right cone.
24.
the locus of all points in a plane such that the sum of the distance from two given points in the plane, called foci, is constant.
25.
the endpoint of each axis.
26.
the sum of whose distances from two fixed points.
27.
the locus of all points in the plane such that the absolute value of the difference of the distances from two given points in the plane, called the foci, is constant.
Down
2.
a hyperbola with perpendicular asymptotes.
3.
the middle.
7.
same distance from a given point.
9.
the axis of symmetry of an ellipse which contains the foci.
10.
the locus of all points in a plane at a given distance, called the radius, from a fixed point on the plane, called the center.
11.
the intersection of a plane with a double right cone resulting in a point, a line, or two intersecting lines.
13.
a line about which a figure is symmetric.
14.
one of the two segments into which the center of an ellipse divides the minor axis.
18.
the ratio of the distance between any point of a conic section and a fixed point to the distance between the same point of the conic section to a fixed line.
20.
the same center.
22.
the common endpoint of two rays forming an angle
Question: What is the ratio of the distance between any point of a conic section and a fixed point to the distance between the same point of the conic section to a fixed line called? Answer: Eccentricity
Question: What is the name of the line that a set of points and only those points that satisfy a given set of conditions lie on? Answer: Graph
Question: What is the center of an ellipse called? Answer: The center | 677.169 | 1 |
Re: Square in a triangle
Re: Square in a triangle
First I was wary about helping, because as you used phrases such as 'as an extension' and 'generalise', it looked suspiciously like something that you had to do for school. However, due to a combination of the facts that it has now been quite a long time, so if it was work you would most probably have handed it in by now, and you gave that cool trigonometry help thingy on the 'This is Cool' board, and it was such an interesting puzzle, I decided to help.
Thinking about the corner opposite the corner of the square that is also in the corner of the triangle, we know that it must have the same x and y co-ordinates (because it's a square) and that the point must lie somewhere on the diagonal line of the triangle (because it's a big square). Using MathsIsFun's graph idea, the equation of the diagonal line in the triangle would be y=-4/3x+4. We also know that the square's opposite corner must lie somewhere on y=x, so we have 2 simultaneous equations.
Substitute in y=x and you get x=-4/3x+4. Add 4/3x: 7/3x=4 Divide by 7/3: x=4/ (7/3)=12/7.
This means that the largest square has a length of 12/7 units and an area of 144/49, or 2 44/49, square units.
Unfortunately, this post is in the realm of the chocolate teapot until someone proves that the biggest square must have the same corner as the triangle and I'm terrible at proofs. The pigeon-hole principle might do it, but I'm not exactly sure how to use it on this.
Re: Square in a triangle
For the extension, I used advanced calculus that I won't bore you with now (unless someone requests it), but basically, if the triangle has perpendicular sides of lengths a and b, then the biggest rectangle will always have sides of length a/2 and b/2, subject to the proof being proved that I mentioned previously. This obviously means that the rectangle will have an area of ab/4, which means that its area will also be half of that of the triangle.
Re: Square in a triangle
I'm not a student -- at least not in the formal sense -- so no worries there! I just posted it as an interesting puzzle in its own right, and subsequently thought of the generalisation to a rectangle.
The rectangle problem can be solved without calculus. (Anyway, did you really mean to imply that advanced calculus is boring?!)
In your diagram above, if the square has a vertex at (0,0) and side a, by similar triangles, (4-a)/a = 4/3, so a = 12/7.
Question: What is the area of the largest rectangle in a triangle with perpendicular sides of lengths a and b? Answer: The area of the largest rectangle is ab/4.
Question: Which mathematical principle might help prove that the largest square must have the same corner as the triangle? Answer: The pigeon-hole principle.
Question: Can the problem of finding the largest rectangle in a triangle be solved without using calculus? Answer: Yes, it can be solved without calculus. | 677.169 | 1 |
Powder Springs, GA Trigonometry and am still responsible for managing students needs. This includes, homework, needs assessment and communication style. I have worked with over 300 middle and high school students in this topic.
...Trigonometry comes from combination of Greek word "trigon" meaning triangle and "metron" meaning measure. Therefore, trigonometry is the measure of triangle regarding its sides, height, angle and areas. So where do other geometric figures come then?
Question: What is the origin of the word 'trigonometry'? Answer: A combination of Greek words "trigon" meaning triangle and "metron" meaning measure | 677.169 | 1 |
decorating it as irregular shapes. rotations and
they wish. Use reflections of
each student's congruent figures
preferred shape (see Esc her designs
13
to create a class
plan for the etropolis.com/escher/
quilt. (use a
variety of
geometric
shapes).
5. Use physical models to Go to a store and Examine the Fit cubicles into
determine the sum of the each student find Biosphere2 (Oracle office space-whose
interior angles of triangles and 10 examples of A2) for geometric design fits the most
quadrilaterals. triangles or shapes and why this people? Compute
quadrilaterals. was done. angles of cubicles.
Explain how Visit an office at
they are school or in the
packaged, stored community and
and displayed. compare their
Examine the arrangement.
angles work
together
tangrams-to fit
different shapes.
6. Extend understanding of Compare X, Y Play "battleship" or Choose a local job or
coordinate system to include coordinates to other games using business and get
points whose x or y values may longitude and coordinates. Or create economic
be negative numbers. latitude on maps a version of "Where in information from a
and globes. Have the world is Carmen local or state
students find Sandiego?" and move website: plot growth
their own city's students between and decline over the
location and locations using past several months
those of their coordinates. or years on the x and
relatives. y axis.
7. Understand that the measure Have students Have each student Compare jobs that
of an angle is determined by trace one of their
take pictures of use short angles
the degree of rotation of an hands on a piece of
trees at home, at (laser surgery) with
angle side rather than the paper with their
school, or local long rays of angles
length of either side. parks. Computefingers spread apart. (telescopes).
the angles of Measure the largest Compare the size of
various tree and smallest angles angles.
branches, between any 2 fingers
regardless of and compare
their length. differences by gender.
Discuss how finger
length does not impact
angle.
8. Predict what three- Using activity #1 Examine floor plans Compare 2-D city
dimensional object will result above, develop and pictures of the planning maps with
from folding a two-dimensional both 2-D and 3- U.S. Capitol. Divide 3-D CAD-CAM
14
net, then confirm the prediction D representa- into small groups and representations and
by folding the net. tions of kitchen choose 1 aspect to views. Use the 2-D
and food storage. develop as a 3-D versions to try to
model. Then rate the predict the 3-D
Have students U.S. capitol virtual images—how
imagine they are tour and compare. accurate is it?
visiting Aruba
on vacation. Have your partner
Question: What should students do to rate the accuracy of virtual images of the U.S. Capitol? Answer: Students should use the 2-D representations and the 3-D virtual images to rate how accurate the 3-D images are. | 677.169 | 1 |
In an isosceles triangle the base is a whole number and is 4 ft. less than the sum of the two equal sides. The perimeter is a whole number between 0 and 75 feet. Find the possible lengths of the equal sides.
What ive got so far is
4x-4=Perimeter
What i dont get is how im sopposed to get the answer with the imformation given.
all help is greatly appreciated.
tkhunny
08-29-2005, 03:31 PM
Rule #1 - Name Stuff - WRITE DOWN clear and concise definitions.
You have a very mysterions formula, 4x - 4 = Perimeter. What is that? Where did it come from? What is 'x'?
Try this:
x = Length of each of the two equal sides of the isosceles triangle.
Now we know what x is and we can proceed. x won't be anything else, as long as we are working on this problem.
We can imagine this, Perimeter > 0, leading to 4*x - 4 > 0, or 4*x > 4, or x > 1. Now we have the possible solutions narrowed down to {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}
We know the length of the third side is a Whole Number. This makes 2*x - 4 >= 0, or 2*x >= 4, or x >= 2. Well, that didn't help.
That's about it. From the problem statement, there is nothing remaining that we have not utilized. We have a solution set for 'x'. It is very little trouble to calculate the related length of each third side and related perimeter, if you wish to do so, even though the problem statement did not ask for it.
Question: What is the formula given in the text for the perimeter of the triangle? Answer: 4x - 4 = Perimeter | 677.169 | 1 |
What is the ratio of the area of ΔABC to ΔEBD?
Given:
The line passing through E and D is tangent to the circle at D.
AC = AB
CAB is a right angle.How many solutions are there to,
A says)21 solutions.
B says) Nope! That is too high.
C says) I counted them, there are 8.
D says) I like 21 too.
E says) I agree with BWell,yeah. That is the GF that gAr usedYeah, I've seen it. What other wayAnd where can we findDo you see 99999 in that huge number am debuggingHow'd you getOkay
Question: What is the first letter of the person who agrees with A's statement? Answer: D | 677.169 | 1 |
cos(a) sin(b) tan(a) cot(b) b; radians ... The values in the table are those angles of the form n ... This gives the sin, cos and tan of 45°. Here is an equilateral triangle where all sides and all angles are equal (to 60°).
A unit circle chart is a chart or sometimes a table that has the most important values of the six standard trigonometric functions -sine(sin),tangent(tan),cosine(cos),cotangent(cot),cosecant(csc)and secant(sec) in the unit circle.
The first trigonometric table was apparently compiled by Hipparchus, ... (sin, cos, tan, and sometimes cis and their inverses). Most allow a choice of angle measurement methods: degrees, radians, and sometimes grad.
degrees and radians, trig functions, use a calculator: Brad Use a calculator...if your calulator has sin,cos,and tan, you can find all six trig functions. However, you need to know your trig to use your calculator effectively. to answer your question, there really isnt one...ive memorized the ...
I need a really good trig table in radians ... Can anyone help please? Hi, my book only lists certain ones (like 5 radian values) ... As for why you see the trig functions listed in the order of sin, cos, tan, csc, sec, cot, that's because it's usually taught in that order.
Use these triangles to complete the following table. 30° 45° 60° sin : cos : tan . Practice Questions. Question 1. A railway is inclined at an angle of 1° to the horizontal. Find the vertical height climbed when a train travels 1 km up the hill. Question 2. Find ...
TRIG, how to find sin cos tan given a value in radian? JohnR asked 5 years ago; Was originally asked on Yahoo Answers United States. ... ok this is tough to explain with no table. Think of a circle and divide it in four sections.
Radians Drawing Reference angle ! sin" cos" ... 210¡! 225¡! 240¡! 300¡! 315¡! 330¡ Example 6) Calculate each of the following expressions. Do not look at the chart on the previous page as you will not have it in an exam. ... sin"! cos"! tan"! csc"!
i need to produce a table chart of sin cos and tan from angle 1-20 using do-while loop. ... tan() functions take their values as radians not degrees. float z=0; You will need to convert your angle (z) from degrees to radians when passing them to sin() etc..
If you didn't find what you were looking for you can always try Google Search
Add this page to your blog, web, or forum. This will help people know what is What is SIN COS TAN RADIAN CHART
Question: Which trigonometric functions can be found using a calculator that has sin, cos, and tan functions?
Answer: All six standard trigonometric functions (sine, cosine, tangent, cotangent, cosecant, and secant)
Question: If a train travels 1 km up a hill inclined at an angle of 1°, what is the vertical height climbed?
Answer: 0.0175 km (approximately)
Question: Which of the following is NOT a standard trigonometric function?
A Sine (sin)
B Tangent (tan)
C Cosine (cos)
D Square root (√)
Answer: D
Question: What is the tangent of a 60° angle in an equilateral triangle?
Answer: √3 | 677.169 | 1 |
A circle of radius 1 is drawn in the plane. Four non-overlapping circles each of radius 1, are drawn (externally) tangential to the original circle. An angle ã is chosen uniformly at random in the interval [0,2ð). The probability that a half ray drawn from the centre of the original circle at an angle of ã intersects one of the other four circles can be expressed as ab, where a and b are coprime positive integers. What is the value of a+b?
Question: How many circles are drawn in total? Answer: 5 | 677.169 | 1 |
Unit 5 Review Sheet
Logic & Triangles
Learning Goal 1: Students will use logic and reasoning to make conjectures MM1G2 a, b
Knowledge and Understanding:
1. Your teacher writes the following sequence of numbers on the board: 3, 7, 11, and 15. He tells
you that each number in the sequence is 4 more than the number before it. You determine that
the next number in the sequence is 19. What type of reasoning did you use to reach your
conclusion? Why?
2. Which is the converse, inverse and contrapositve of the following statement?
Statement: If a quadrilateral has four congruent sides, then it is a rhombus.
Application:
3. Write the following statement as a biconditional statement?
If a quadrilateral is a square, then it has 4 right angles.
4. Consider the following statements. What is a valid conclusion that can be reached using the
following statements? (Hint: Use the Law of Syllogism/Chain Rule)
Statement: If Takira does not have any homework this weekend, she is going camping with her
friends.
Statement: Takira has homework.
Extension:
5. What additional statement is required to reach the given conclusion? Why?
Statement: Students can get free admission to the movie if they present their student
identification card.
Statement: ?
Conclusion: Therefore, Mary did not bring her student identification card.
Learning Goal 2: Students will determine possible lengths of triangles as well as determine if two
triangles are congruent. MM1G3 b, c Score:
Knowledge and Understanding D M E
1. The sides of a triangle are 12, 7, and x. Use the Triangle Inequality Theorem to determine the
length of the third side.
2. Use the diagram below to complete the congruence statements ∆GJH ∆_____ by ______.
H
G J I
Application: G
3. In ∆CAT, m < C = 540 and m < T = 360. What is the longest side of the triangle? (Hint: Draw
a picture)
4. What information would be needed in order to prove the following two triangles congruent by
SAS?
K L
N
J M
Extension:
5. Using the diagram below, order the sides from greatest to least.
d
40 0 450
280
e
h g
280
f
Learning Goal 3: Students will understand and use points of concurrency in a triangle. MM1G3 e
Score:
Knowledge and Understanding
D M E
1. The medians of a triangle concur at the _____________________.
2. The Circumcenter is the point of concurrency in a triangle where the __________________
intersect.
Application:
Question: What information would be needed to prove two triangles congruent by SAS? Answer: The lengths of two pairs of corresponding sides and the included angle between those sides.
Question: What additional statement is needed to reach the conclusion "Therefore, Mary did not bring her student identification card"? Answer: "Mary does not have any homework this weekend."
Question: In the diagram below, order the sides from greatest to least: d, e, f, g, h. Answer: h, g, e, d, f | 677.169 | 1 |
(e) A logarithmic function
31. Refer again to Fig. T-0-2. The coordinate system in this illustration is
(a) polar.
(b) spherical.
(c) semilog.
Fig. T-0-2 Illustration for Part Zero Test Questions 30 and 31.
(d) log-log.
(e) trigonometric.
32. Suppose that there are two vectors. Vector a points straight up with a magnitude of 3, and vector b points directly toward the western horizon with a magnitude of 4. The cross product a × b has the following characteristics:
(a) It is a scalar with a value of 12.
(b) It is a vector pointing toward the southern horizon with a magnitude of 12.
(c) It is a vector pointing upward and toward the west with a magnitude of 5.
(d) It is a vector pointing straight down with a magnitude of 5.
(e) We need more information to answer this.
33. Using a calculator, you determine the power of 2 (that is, 2 ) to four significant figures. The result is
(a) 1.587.
(b) 2.828.
(c) 4.000.
(d) 8.000.
(e) The expression 2 is not defined and cannot be determined by any means.
34. The numbers 34 and 34,000 differ by
(a) a factor of 10.
(b) three orders of magnitude.
(c) five orders of magnitude.
(d) seven orders of magnitude.
(e) the same ratio as a foot to a mile.
35. Right ascension is measured in
(a) degrees.
(b) radians.
(c) linear units.
(d) logarithmic units.
(e) hours.
36. Consider the function y = 2 x with the domain restricted to 0 < x < 2. What is the range?
(a) 0 < y < 1/2
(b) 0 < y < 1
(c) 0 < y < 2
(d) 0 < y < 4
(e) There is not enough information given to answer this question.
37. The fifth root of 12 can be written as
(a) 12 1/5 .
(b) 12/5.
(c) 12 5 .
(d) 5 12 .
(e) 5 1/12 .
38. Suppose that you are given the equation x2 + y2 = 10. What does this look like when graphed in rectangular coordinates?
(a) A straight line
(b) A parabola
(c) An elongated ellipse
(d) A hyperbola
(e) A circle
Question: Is the cross product of vectors a and b a scalar? Answer: No | 677.169 | 1 |
We can draw a right triangle with an angle of measure x and a hypotenuse of length 35. If we label the side opposite the angle x with variable O, and the side adjacent to (next to) angle x by A, then we get the picture in Figure 5.32.
We know that sin(x) = = 0.73, so 0 = 35 · (0.73) = 25.55.
To find the length of the adjacent side, we use the Pythagorean theorem:
A2 + (25.55)2 = 352
A = √572.1975 ≈ 23.92
Thus, the lengths of the other two sides are about 25.55 feet and 23.92 feet.
Question: What is the relationship between the lengths of the sides of the triangle? (Choose the correct option)
A) A < O
B) A = O
C) A > O
Answer: C | 677.169 | 1 |
I've already got a test to determine that the line segment does not intersect with the cube, but now I need to determine the distance. I've got a few ideas, but I'm looking for the fastest known solution. Any suggestions?
Two things are infinite: the universe and human stupidity; and Im not sure about the universe. -- Albert Einstein
Question: Which of the following is NOT mentioned in the text: a) a test, b) a cube, c) a solution, d) a car? Answer: d) a car | 677.169 | 1 |
Common Core Standards: Math
Math.G-CO.6
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
The words "rigid" and "motion" sound like complete opposites, don't they? Well, not in geometry. A rigid motion is a motion—or transformation, in geometric lingo—that preserves distance and lengths; in other words, every segment of the image is congruent to its preimage.
Students should already be comfortable with the ideas of translations, reflections, and rotations. They have to use rigid motions to not only transform figures and predict the effect it has on a specific figure, but also to determine when two figures are congruent.
Congruence between two geometric objects can be defined as a rigid motion on the coordinate plane that maps one object onto another. Students should be able to recite this this definition verbatim. Or, more importantly, they should actually understand it.
The key here is to differentiate between congruence and carrying one object onto another. While two reflected objects can't always be carried onto one another, they can still be congruent because all that separates them is reflection—a rigid motion. Make sense?
Once students fully understand how these topics fit together, they will be able to use transformations to not only find and prove properties of geometry, but to create patterns, including tessellations.
If your students are giving you a hard time with why these things are important, remind them that if they ever want to dabble in design, whether it's fashion, interior, or video game, they have to identify and create patterns. At the root of patterns is, you guessed it, geometry. That'll get them thinking about transformations and congruence the next time they watch Project Runway or play Call of Duty.
Drills
There are three basic types of transformations: translation, rotation, and reflection. But there are four types of rigid transformations – translation, rotation, reflection, and one that combines two of these. Name and define the fourth one.
(A) Reflection translation: a reflection over a line, then a translation in the same direction as that line(B) Reflection rotation: a reflection over a line, then a rotation to bring the image to the same position as its original image(C) Rotation translation: a rotation of minimum order to bring the image to its original positioning then a translation over a line in either direction(D) Glide reflection: a reflection over a line, then a translation in the same direction at that line
Correct Answer:
Glide reflection: a reflection over a line, then a translation in the same direction as that line
Answer Explanation:
It is a bit deceiving, given that the name does not entirely "reflect" its true meaning; however, to transform the image, it is first reflected over a line then translated in the same direction as that line of reflection. Although (A) sounds more logical for this, the terms are incorrect. Rotation isn't involved in the fourth rigid motion, which makes (B) and (C) incorrect.
Question: What are the four types of rigid transformations? Answer: Translation, rotation, reflection, and glide reflection.
Question: How can students determine if two figures are congruent? Answer: By using the definition of congruence in terms of rigid motions, which maps one object onto another.
Question: What is a glide reflection? Answer: A glide reflection is a reflection over a line, then a translation in the same direction as that line. | 677.169 | 1 |
A triangle is dilated by 2, translated five units to the right, and then compressed by 2. Which of the following is true?
(A) The transformations are rigid because the final triangle is congruent to the original triangle(B) The transformations are non-rigid because the final triangle is not congruent to the original triangle(C) The transformations are rigid because the triangle stays the same size throughout(D) The transformations are non-rigid because the triangle does not stay the same size throughout
Correct Answer:
The transformations are non-rigid because the triangle does not stay the same size throughout
Answer Explanation:
Although the final triangle and the original triangle are congruent, the transformations performed aren't rigid because they don't maintain congruency throughout. Dilation increases the size of the image and compression decreases it. The triangle grows and shrinks by the same amount, but it doesn't change the fact that its size still changes.
A square mirror has a horizontal scratch in the bottom right corner. If you reflect the mirror across a horizontal axis and then rotate it 90° counterclockwise, where will the scratch be and how will it be oriented?
(A) Vertically in the bottom right corner(B) Horizontally in the top right corner(C) Vertically in the top left corner(D) Horizontally in the bottom left corner
Correct Answer:
Vertically in the top left corner
Answer Explanation:
If we reflect the mirror across a horizontal axis, the scratch will end up in the top right corner, still horizontal. Then, rotating the mirror 90° counterclockwise places the scratch in the top left corner. Rotating a horizontal line 90°, though, means it's changed from horizontal to vertical. That means our answer is (C). If you aren't sure, scratch a mirror and try it yourself
Question: How is the scratch oriented after the transformations? Answer: Vertically
Question: Are the transformations rigid or non-rigid? Answer: Non-rigid | 677.169 | 1 |
The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs. So in the figure above the angle would be ABC or CBA. So long as the vertex is the middle letter, the order is not important. — "Angle definition - Math Open Reference",",
GOP Senate candidate Sharron Angle, who lost her bid to unseat Senate Majority Leader Harry Reid (D-Nev. could be back on the campaign trail in 2012. — "Angle considering 2012 run for Congress - The Hill's Ballot Box",
FALLON, Nev. — Sharron Angle, who recently lost a bid to unseat Sen. Harry Reid, said on Saturday she still has a strong desire to serve her f (read more). — "Angle plans to continue political career | ",: Shocked by loss, has political 'options' - Politics", tri-
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select save picture as to save it on your disk Option 2 Select set as background to directly set the picture as your desktop wallpaper Detailed instructions faq
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Not so much a review but a list of features of Pioneers MEP 7000 IMG http www pioneer eu images products cdplayerdj pioneer mep7000 angle medium jpg The MEP or Multi Entertainment Player and Controller is aimed at the mobile DJ and bar market
the same function f T T where T is the upper temperature of heating at which the sample can return to diamond after cooling You can see the radial and the angle distribution functions here See movie picture
when the poodle fur is done flying expect Hughes physical strength to turn out Lytle s lights In addition to this impending card the fight industry has been abuzz with word that 1996 Olympic heavyweight gold medalist Kurt Angle plans to make his mixed martial arts debut sometime within the next year Angle who would need to pass a physical to assure that
the web for banshee pics ok and i came across this little cartoon pic of him so i was like woohoooo he is good so from then on i was looking at them finally i have tryed one and here is angle i tryed my hardest so could some one please give me there thoughts and please if u do these micro heros feel free to show them off
Question: What does the angle symbol represent in mathematics? Answer: The angle symbol represents an angle in mathematics, with the middle letter being the vertex and the other two on the legs.
Question: What is the function 'f' in the context of materials science? Answer: 'f' is the radial and angle distribution functions in the context of materials science.
Question: What is the name of the cartoon character the user found pictures of? Answer: Banshee | 677.169 | 1 |
FORMATIVE: Pretest: Lesson 1 used to determine current level of knowledge and understanding of geometry concepts. Triple-Entry Journal: Lesson 2, 3 (record vocabulary), lesson 5 (students will record vocabulary associated with the Pythagorean Theorem), lesson 7 (students will demonstrate their thinking as they develop an aquatic habitat). Quick Write: Lesson 3 (record students' process for applying learning about dilating an object), lesson 5 (students will share what they know about right angles). Build Models: Lesson 5 (students will create a concrete model of the Pythagorean Theorem), lesson 7 (students will design and illustrate geometric habitats). Think-Pair-Share: Lesson 2 (activate prior knowledge), lesson 6 (students will recall elements of and process for using the Pythagorean Theorem). Give One, Get One: Lesson 2 (solidify concepts or future application), lesson 6 (review unit to date and prepare for geometry project). Knowledge Rating Guide: Lesson 4 (gage students' current level of understanding and compare following the lesson). Two-Column Notes: Lesson 4 (journal size paper with lines formatted in four rows and students will record their understanding of angle measures and rules for finding them). Creative Writing Prompt: Lesson 5 (students will express their understanding of the importance of the radical, what its function is). Problematic Situation: Lesson 6 (students will problem solve using the Pythagorean Theorem). Coding: Lesson 5 (students will read about the Pythagorean Theorem and code for J+, put in journal because it's new, C-I made a connection, ? – I don't understand/I wonder, checkmark – I know this). Museum Walk: Lesson 7 (students will ask questions about peers aquatic designs). Journal writing: Lesson 7 (students will respond to at least two questions about their design posed by their peers).
ACTIVE LEARNING EVENTS: Students will activate prior knowledge using a matching up activity for formulas and shapes. Students will record definition and formulas in their journals. Students will transform two dimensional objects on a coordinate graph. Students will use tools to create parallel lines and transversals. Students will find the measure of all angles given the measure of one angle when working with parallel lines with a transversal. Students will prove the Pythagorean Theorem using graph paper and a right triangle. Students will use the Pythagorean Theorem to form two lengths into a right angle. Students will demonstrate their understanding of geometry by designing and problem solving an aquatic habitat for a pair of sea creatures.
This unit engages students in an in-depth study of the characteristics of two dimensional and three dimensional shapes. It is designed to help students understand the differences among area, surface area and volume. Students will explore the building blocks of shapes and will solve real world problems related to volume and surface area. The instructor uses Give One, Get One, RAFT, Semantic Feature Analysis, Frayer Model, Think-Pair-Share, Triple-Entry Journals, and Word Walls. Technology integration includes the use of an interactive white board, document camera, virtual manipulatives, and OER resources.
Question: What will students do in the "Creative Writing Prompt" for lesson 5? Answer: Students will express their understanding of the importance of the radical and its function.
Question: What is the purpose of the "FORMATIVE" section in the text? Answer: The "FORMATIVE" section is a pretest used to determine the current level of knowledge and understanding of geometry concepts.
Question: What is the main activity in "Build Models" for lesson 5? Answer: Students will create a concrete model of the Pythagorean Theorem.
Question: In which lesson will students prove the Pythagorean Theorem using graph paper and a right triangle? Answer: Lesson 6 | 677.169 | 1 |
Universal parabolic constant
It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter. It is denoted P.[1][2][3] In the diagram, the latus rectum is pictured in blue, the parabolic segment that it forms in red and the focal parameter in green. (The focus of the parabola is the point F and the directrix is the line L.)
The value of P is
(sequence A103710 in OEIS). The circle and parabola are unique among conic sections in that they have a universal constant. The analogous ratios for ellipses and hyperbolas depend on their eccentricities. This means that all circles are similar and all parabolas are similar, whereas ellipses and hyperbolas are not.
Question: What is the ratio that the universal parabolic constant represents? Answer: The arc length of the parabolic segment formed by the latus rectum to the focal parameter | 677.169 | 1 |
non-Euclidean
[nän′yo̵̅o̅ klid′ē ən]
adjective
designating or of a geometry that rejects any of the postulates of Euclidean geometry, esp. the postulate that through a given point only one line can be drawn parallel to another line that does not contain the given point
non-euclidean - Science Definition
Relating to any of several modern geometries that are based on a set of postulates other than the set proposed by Euclid, especially one in which all of the postulates of Euclidean geometry hold except the parallel postulate . Compare Euclidean.
Question: Can you compare the nature of parallel lines in Euclidean and non-Euclidean geometries? Answer: In Euclidean geometry, through a given point, only one line can be drawn parallel to another line. In non-Euclidean geometry, this is not necessarily the case, and more than one line can be drawn parallel to another line. | 677.169 | 1 |
The Pythagorean Identities get their name because they are based on the famous Theorem of Pythagoras. You are very likely already familiar with it. Simply, for a right angle triangle, it says "the square of the hypotenuse is the sum of the squares of the other two sides." Mathematically, you have seen this represented as:
a2 + b2 = c2, where a and b are sides and c is the hypotenuse.
Now, I will show you how to derive these special trig identities, using this theorem as our starting point. To do this, we need to start with a right triangle, created by the radius of a unit circle and the axis:
How to label a right triangle inscribed inside a unit circle
We can say that the right triangle formed by dropping a line from the point that the radius touches the circle down to the axis has a base of x units long and y units high. The radius in a unit circle, by definition, is 1. Now, let's apply the definitions of sine and cosine to our triangle. Recall:
sin(ɵ) = opposite / hypotenuse = y / 1 = y
cos(ɵ) = adjacent / hypotenuse = x / 1 = x
So, now we can relabel our diagram by substituting in these basic trig identities.
Right angle triangle with sides x and y renamed
With the triangle now correctly labeled for our derivation, we can apply the Theorem of Pythagoras to arrive at one of the Pythagorean Identities. Since a2 + b2 = c2, we can therefore equate the sides of our triangle to these terms to give us our first of the trig Pythagorean Identities:
sin2(ɵ) + cos2(ɵ) = 1
If you have followed along up till now and understood everything I've done, then you are well on your way to remembering this trigonometric identity. If you can remember how to derive you, you don't even have to memorize it (though it always helps!) For the next Pythagorean Identity, you start with this first identity, and you apply some basic algebra and trigonometry to it to derive the second and third identities. Recall the definitions of secant, cosecant, and cotangent:
sec(ɵ) = 1 / cos(ɵ)
csc(ɵ) = 1 / sin(ɵ)
cot(ɵ) = 1 / tan(ɵ) = cos(ɵ) / sin(ɵ)
With those inverse trig functions in mind, let's take the first Pythagorean Identity and divide all of its terms by cos2(ɵ). That gives you:
1 / cos2(ɵ) = sin2(ɵ) / cos2(ɵ) + cos2(ɵ) / cos2(ɵ)
sec2(ɵ) = tan2(ɵ) + 1
Question: What is the second Pythagorean Identity derived from the first one? Answer: sec²(θ) = tan²(θ) + 1
Question: If the angle θ is 45 degrees, what is the value of sin²(θ) + cos²(θ)? Answer: 1 (since sin(45°) = cos(45°) = √2/2) | 677.169 | 1 |
Topic review (newest first)
Very good. Did you draw a diagram on that trig problem about the house?
mathstudent2000
2013-09-03 03:48:22
thanks
bobbym
2013-09-02 13:14:09
Forcing them to computational math would reduce their numbers real quick.
anonimnystefy
2013-09-02 12:26:51
Unfortunately, that does not reduce their numbers. Only one thing does.
bobbym
2013-09-02 09:41:19
I think we should consider all topologists the same.
anonimnystefy
2013-09-02 09:36:40
Definitely! And there's so many of 'em!
bobbym
2013-09-02 09:20:29
Yea, they are weird.
anonimnystefy
2013-09-02 09:19:07
And, it would make even topologists happy. They already think all triangles are the same!
bobbym
2013-09-02 09:18:05
All triangles should be named ABC by law. What if you have a bunch of them? No difference!
anonimnystefy
2013-09-02 09:15:15
Hi bobbym
My bad! Thought it was GIH again. Your answer is correct for 5!
bobbym
2013-09-02 09:06:17
Hi;
anonimnystefy
2013-09-02 08:59:17
Hi
bobbym
2013-09-02 08:57:38
Hi;
mathstudent2000
2013-09-02 08:05:10
1. What is the sine of an acute angle whose cosine is 7/25?
2. I'm standing at 300 feet from the base of a very tall building. The building is on a slight hill, so that when I look straight ahead, I am staring at the base of the building. When I look upward at an angle of 54 degrees, I am looking at the top of the building. To the nearest foot, how many feet tall is the building?
3. If A is an acute angle such that \tan A + \sec A = 2, then find \cos A.
4. In triangle GHI, we have GH = HI = 25 and GI = 30. What is \sin\angle GIH?
Question: What was the last message in the conversation? Answer: Very good. Did you draw a diagram on that trig problem about the house?
Question: Who is the user who asked about the trigonometry problem involving a house? Answer: mathstudent2000 | 677.169 | 1 |
Every body has only one point at which whole the weight of body can be supposed to be concentrated, this is called center of gravity (C.G.) of the body.
For plane figures (like circle, rectangle, quadrilateral, triangle etc.), which have only areas and no mass, the point at which the total area of plane is assumed to be concentrated, is known as the controid of that area.
The center of gravity and centroid are at the same point for plane body.
Determination of centroid of an area: The centroid of an area is defined by the equations.
x ¯ = ∫ x dA/A, y ~= ∫ t dA/A
Centroid = (x ¯,y ¯) on x-y plane.
where A denotes the area. For a composite plane area, composed on N subareas Ai, each of whose centroidal coordinates xi ¯ and y ¯i are known. The integral is replaced by summation.
Centroid of few common figures are given in Table.
Table
Table
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Related Questions
Question: Which of the following is NOT a plane figure? A) Circle B) Sphere C) Rectangle D) Triangle Answer: B) Sphere (as it is a solid figure)
Question: What is the point at which the entire weight of a body can be considered to be concentrated? Answer: The center of gravity (C.G.) | 677.169 | 1 |
54x12
What is the length of the hypotenuse of a right triangle with legs that are 7 and 24 inches long, respectively?
2
7
25
31
Kayla purchased a new phone at 31% off its original price. If she paid $34 for the phone, which of the following is an equation that could be used to find x, the original price of the phone?
34 = 1.31x
31 = x – 0.34x
31 = x – 0.66x
34 = x – 0.31x
x = x – 0.31
What are the values for x that satisfy the equation x2 – 5x – 14 = 0 ?
–2 and 9
–2 and –9
2
2 and –7
–2 and 7
Medium
In the standard (x, y) coordinate plane, the midpoint of is (–3, 5). If point M is located at (2, 3), what are the coordinates of point N ?
(–8, 7)
(7, 1)
(–1,8)
(–6, 15)
(0, 0)
For right triangle ΔABC, the sides measure 6, 8, and 10 inches. Which of the following equations could be used to solve for smallest angle, x, which is opposite the smallest side?
sin x° = 0.6
sin x° = 0.8
tan x° = 0.8
sin x° = 0.75
tan x° = 0.6
If m < 0, which of the following is greatest?
m
3 m
6 m
9 m
It cannot be determined from the information given.
If triangle MNO is similar to triangle PQR shown below, which of the following could be the measures of the three sides of triangle MNO ?
If 4 times a number p is subtracted from 20, the result is negative. Which of the following gives the possible value(s) for p ?
0 only
5 only
20 only
All p < 5
All p > 5
Which of the following could be the equation of the graph below?
y = x – 1
y = x2 –1
y = (x – 1)2
y = (x +1)2
y = x2
The inequality 3(3x – 2) < 6 – (5x – 2) is equivalent to
x < 7
x < 5
x < 3
x < 1
x < –1
Hard
Question: If triangle MNO is similar to triangle PQR with sides in the ratio 2:3:4, which of the following could be the measures of the three sides of triangle MNO? Answer: 4, 6, 8
Question: If m < 0, which of the following is greatest? Answer: 9 m | 677.169 | 1 |
A mnemonic to help you remember:
The C in Complementary stands for Corner, 90˚
The S in Supplementary stands for Straight, 180˚
Have a look at the following videos for further explainations of complementary angles and supplementary angles:
How to identify and differentiate complementary and supplementary angles.
This video describes complementary and supplementary angles with a few example problems. It will also explain a neat trick to remember the difference between complementary and supplementary angles.
How to Find the Measure of Complementary Angles Using Algebra
Complementary & Supplementary Angles Word Problem
Complementary Word Problem
How to solve a word problem about its angle and its complement
The measure of an angle is 43 more than its complement. Find the measure of each angle.
Complementary and Supplementary Angles - Example 1
What it means for angles to be complementary and supplementary and do a few problems to find complements and supplements for different angles.
Complementary and Supplementary Angles - Example 2
Create a system of linear equations to find the measure of an angle knowing information about its complement and supplement.
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Question: What does the S in Supplementary stand for? Answer: Straight | 677.169 | 1 |
The exterior angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to two right angles.
2. c) Practice Proposition 32.
13. Prove Proposition 32 by drawing a straight line DE through A 13. parallel to BC.
3. This proof is attributed to Pythagoras, who lived some 250 years 3. before Euclid.
Because DE is parallel to BC,
the angle DAB is equal to ABC,
because they are alternate angles;
for the same reason, angle EAC is equal to ACB.
Angles DAB, BAC, CAE are together equal to two right angles. (I. 13)
Therefore the three angles of the triangle, ABC, BCA, CAB, are together equal to two right angles.
14. ABC is a circle with center D; ABD is a triangle; and ADC is a
14. straight line. Prove that angle BDC is double angle A.
AD is equal to DB because they are radii of the circle;
therefore angle B is equal to angle A. (I. 5)
The exterior angle BDC is equal to the two opposite interior angles together,
angles A and B; (I. 32);
therefore, angle BDC is double angle A.
The three angles of a triangle are equal to two right angles,
and the right angle itself accounts for one of them;
therefore, the remaining two angles together must equal one right angle.
This also implies that each of them must be less than a right angle, an acute angle.
Question: What is the sum of the three interior angles of a triangle? Answer: Two right angles
Question: What type of angles are the two remaining angles in a triangle, after accounting for the right angle? Answer: Acute angles | 677.169 | 1 |
Hal agrees that the teacher is illustrating his idea. Some students initially disagree that this idea, two circles in different planes, could be possible. In the end, the teacher and students accept Hal's idea because the question does not state that the circles must be in the same plane. Through discussing Hal's question, the students move beyond the assumption that their circles must lie in the same plane. Mathematicians know through experience that they must not make unfounded assumptions when solving a problem.
In the following excerpt, we can see that Hal believes that his thinking has value. We can also see that Hal's idea opens up the discussion.
Teacher. Hal has thought of an example of two circles that do have the same center. One circle is headed in this direction in one plane and one, in this …
The rest of this article is only available to active members of Questia
Question: What did Hal believe about his thinking? Answer: Hal believed that his thinking had value. | 677.169 | 1 |
Reviewing the Distance Formula between points on the Cartesian Coordinate System
To review the Distance Formula between two points on the Cartesian Coordinate System (on the x-y coordinate system):
The "Distance Formula" tells us that the difference between two y-points squared added to the difference between two x-points squared will give us the square of the distance between x and y.
The formula tells us that if you get the difference between the 2 y-values and square that difference, get the difference between the 2 x-values and square that difference, then add these two differences together and finally, take their square root, you
would arrive at the distance between the two points.
The distance formula looks like this: d-squared = (the difference in the y's squared) + (the difference in the x's squared).
Now it is beginning to look like the Pythagorean theorem of c squared = a squared + b squared.
The Pythagorean theorem of geometry states:
The square of the length of the hypotenuse of a right triangle = the sum of the squares of the lengths of the two shorter sides of the right triangle.
Or more succinctly stated: c squared = a squared + b squared. (c being the hypotenuse of a right triangle, a and b being the 2 shorter legs of that same right triangle.)
So, the distance formula is derived from the Pythagorean theorem. This makes it easier to remember, if we are familiar with the Pythagorean theorem.
Question: What does 'd' represent in the distance formula? Answer: The distance between the two points
Question: If the coordinates of two points are (3, 4) and (6, 8), what is the distance between them? Answer: 5 | 677.169 | 1 |
Euclid's argument is as follows:
Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, namely AB equal to DE and AC equal to DF, and the angle BAC equal to the angle EDF.
I say that the base BC also equals the base EF, the triangle ABC equals the triangle DEF, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE.
If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE.
Russell has a problem with superimposing the two triangles. Are these material triangles or abstract triangles? If they are material, they cannot be perfectly rigid. In addition when superimposed, they are bound to be slightly off.
Russell proves that with the addition of another (kind of complicated) axiom, this theorem is still true.
Book I, Proposition XII (To draw a straight line perpendicular to a given infinite straight line from a given point not on it.)
Here is Euclid's original argument.
Let AB be the given infinite straight line, and C the given point which is not on it.
It is required to draw a straight line perpendicular to the given infinite straight line AB from the given point C which is not on it.
Take an arbitrary point D on the other side of the straight line AB, and describe the circle EFG with center C and radius CD. Bisect the straight line EG at H, and join the straight lines CG, CH, and CE.
I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it
Since GH equals HE, and HC is common, therefore the two sides GH and HC equal the two sides EH and HC respectively, and the base CG equals the base CE. Therefore the angle CHG equals the angle EHC, and they are adjacent angles.
But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.
Russell claims that the existence of H required an axiom of continuity.
Question: According to Euclid, what proves that CH is perpendicular to AB? Answer: The fact that the adjacent angles CHG and EHC are equal, and when a straight line standing on another makes adjacent angles equal, each of the equal angles is right, proving CH is perpendicular to AB.
Question: What does Russell claim regarding the existence of point H in Euclid's proof? Answer: Russell claims that the existence of point H required an axiom of continuity.
Question: According to Euclid, what happens when triangle ABC is superposed on triangle DEF? Answer: The point B coincides with E because AB equals DE. | 677.169 | 1 |
The Fixed Point Theorem
Question #1: I climbed a mountain, following a trail, in six
hours (noon to 6 PM). I camped on top overnight. Then at noon the next day, I
started descending. The descent was easier, and I made much better time. After
an hour, I noticed that my compass was missing, and I turned around and
ascended a short distance, where I found my compass. I sat on a rock to admire
the view. Then I descended the rest of the way. The entire descent took four
hours (noon to 4 PM). I thought that I remembered that there was a place on the
trail where I was at the same place at the same time on both days. Can you tell
if I was right?
Answer #1: Yes, there must be one "fixed point"
on my trip. Here is a graph showing my height on the mountain versus time.
Regardless of the changes in rate of travel, it should be obvious that the two
paths must cross at one point, at least.
The problem was stated so that one path could not go around the end of
the other path. This situation is ignored by Euclid, but modern geometry text
books give this postulate, or something like it: If two points are on either
side of another line, then a line drawn through the two points intersects the
other line. This is true of curved line segments as well as infinite
straight lines, as long as we are careful to prevent a line from going around
the end of the other line.
The above situation is an example of the fixed point theorem, which
states (according to MathWorld):
"If g is a continuous function g(x) is an element of [a,b] for all x elements
of [a,b], then g has a fixed point in [a,b]." Further, a fixed point is "a
point which does not change upon application of a map, system of differential
equations, etc." Above we are actually applying a map. The fixed point theorem
has applications in almost all branches of mathematics.
Question #2: I have two maps of the USA (not the same scale). I
crumple one of them up into a loose ball and place it on top of the other map
entirely within the borders of the USA on the flat map. I wonder if there is
any point on the crumpled map (that represents the same place in the USA on
both maps) that is directly over it's twin on the flat map. What do you
think?
Answer #2: Of course the point we are wondering about is a fixed
point. Yes there must be at least one point on both maps that remains fixed
(directly above or under its twin on the other map).
Question: What is the graph showing in the text? Answer: The graph shows the hiker's height on the mountain versus time.
Question: Which mathematical branches does the Fixed Point Theorem have applications in? Answer: Almost all branches of mathematics.
Question: What is a fixed point in the context of the theorem? Answer: A point which does not change upon application of the function. | 677.169 | 1 |
The usual reference to the Problem of Apollonius is to construct a circle tangent to three given circles. The history
of mathematics (for example, see Heath (1981, Vol. II, pp. 181-182))
indicates that Apollonius did in fact devote Book II of his treatise
to this construction. Apollonius did, however, have a lot of developmental
material in Book I. Pappus (Who was Pappus?)
was credited with this general statement of the problem:
Given three things, each of which may be
either a point, a straight line, or a circle, to draw a circle
which shall pass through each of the given points (so far as
it is points that are given) and touch the straight lines or
circles.
The problem of most interest, of course, is
the case where the three objects are circles. Plane geometry methods
of the three circle problem usually make use of being able to
do simpler constructions such as a circle tangent to two given
circles and through a given point. Such was the case of Apollonius
in which he built on the work of Euclid (Euclid had presented
the constructions for three points and for three lines). Apollonius
presented a construction for all of the other cases in Book I before
completing the construction for the three circle case in Book
II. This was also the approach of Altshiller-Court (1925) where
he gave the construction of CCC (Section 355) by reducing it to
the construction of PCC (Section 352). The PCC construction was
reduced to the PPC construction (Section 351). The development
of the entire set of constructions is given in Sections 343 to
355.
We will re-phrase this as: To construct a circle
tangent to three given objects -- points, lines, or circles.
A point is "tangent to" a circle
if it is on the circle.
The problem is posed for "general position" -- that
is, objects arranged so that constructions are possible.
There are 10 classes of constructions for this problem.
1. 3 points (PPP)
2. 2 points and 1 line (PPL)
3. 2 points and 1 circle (PPC)
4. 1 point, 1 line, and 1 circle (PLC)
5. 1 point and 2 lines (PLL)
6. 1 point and 2 circles (PCC)
7. 1 line and 2 circles (LCC)
8. 2 lines and 1 circle (LLC)
9. 3 lines (LLL)
10. 3 circles. (CCC)
Generate constructions for items
1 - 9 above.
PPP: Given three points, construct a circle
through the three given points. Clear. It is the circle through
Question: Who was Apollonius? Answer: Apollonius was a Greek mathematician who lived in the 3rd century BCE and is known for his work on conic sections.
Question: What did Pappus state as the general problem? Answer: Pappus stated the general problem as: Given three things, each of which may be either a point, a straight line, or a circle, to draw a circle which shall pass through each of the given points and touch the straight lines or circles.
Question: Which mathematician presented a construction for all of the other cases before completing the construction for the three circle case? Answer: Apollonius
Question: What is the Problem of Apollonius? Answer: The Problem of Apollonius is the construction of a circle tangent to three given circles. | 677.169 | 1 |
The Application of Trigonometry
Please view the attached file to see the diagram which accompanies this question.
1. Find the length L from point A to the top of the pole.
2. Lookout station A is 15 km west of station B. The bearing from A to a fire directly south of B is S 37°50' E. How far is the fire from B?
3. The wheels of a car have a 24-in. diameter. When the car is being driven so that the wheels make 10 revolutions per second, how far with the car travel in one minute?
4. A regular octagon is inscribed in a circle of radius 15.8 cm. Find the perimeter of the octagon.
5. What is the angle of elevation of the sun when a 35-ft mast casts a 20-ft shadow?
6. A V-gauge is used to find the diameters of pipes. In the figure on p. 373 in the text, the measure of angle AVB is 54°. A pipe is placed in the V-shaped slot and the distance VP is used to predict the diameter.
a. Suppose that the diameter of a pipe is 2 cm. What is the distance VP?
b. Suppose that the distance VP is 3.93 cm. What is the diameter of the pipe?
c. Find the formula for d in terms of VP.
d. Find a formula for VP in terms of d.
The line VP is calibrated by listing the corresponding diameters as its units. This, in effect, establishes a function between VP and d.
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Solution Summary
This solution is comprised of detailed explanations of the application of the trigonometric functions such as sine, cosine, and tangent. With diagrams and step-by-step contents, the solution should provide the students a clear understanding of trigonometric functions in real life.
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Question: What is the length L from point A to the top of the pole? Answer: This information is not provided in the given text. | 677.169 | 1 |
Learn more: Multiple proofs showing that a point is on a perpendicular bisector of a segment if and only if it is equidistant from the endpoints. Using this to establish the circumcenter, circumradius, and circumcircle for a triangle
Learn more: Showing that three points uniquely define a circle and that the center of a circle is the circumcenter for any triangle that the circle is circumscribed about
Learn more: Thinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector
Learn more: Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter).
Question: Is it true that a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints? Answer: Yes | 677.169 | 1 |
It's pretty obvious that the points where the circles intersect are at integer multiples of some constant distance from the foci. What's not so obvious (and what I just recently realized thanks to your circle-and-line moirees) is that those points of intersection (along one particular ellipse) also are equally-spaced in the direction of the major axis - in this case, the X direction. To me, that's a surprising observation.
Through my circle and line moires you have evidently become aware of the Directrix Line.
This line is perpendicular to the major axis. The ratio of a conic section point's distance from the focus to its distance from the directrix is a constant. This constant is the conic section's eccentricity.
Here are some nice illustrations of a conic section's directrix line: [link]
I also did a more elaborate Dandelin page which talks about the directrix line (among other things): [link]
I was already aware of it in the case of parabolas. Somewhat recently, I've heard it referenced for other conics on Wikipedia, but your illustrations definitely helped my understanding.
There's another concept I've been considering lately. It's similar to the directrix except the ratio of distances is always one. In the case of a parabola, it's the directrix. For an ellipse, it's not a line at all, but a circle which encloses the focus (and the ellipse). (That case comes up in the study of the Apollonian problem). For a hyperbola, it's a circle not enclosing the focus. In both cases, the center of the circle is the other focus. Is there a name for this circle?
This is something new to me! If you find anything online about this, please send me a link.
Did a quick and dirty illustration of the ellipse you describe: [link] Haven't measured line segments, but they look right.
Just thought of a demonstration that convinces me it's true (don't know if you'd call it a proof). Light rays emanating from one focus will reflect to the next focus. Add the length of a reflected light ray to the original ray and you will have the length of the ellipse's major axis. The radius of this circle is the Ellipse's major axis.
Yeah, you pretty much nailed it with that illustration. And the concept isn't particularly inventive either.
Here's the connection to stuff I've seen before: The points on the ellipse are the centers of possible circles that are tangent to the circle and intersect the point. If you replace the point with a circle of some comparatively small radius, and shrink the bigger circle by the radius of the smaller circle, the family of circles tangent to both those circles have centers along that same ellipse again. (See Steiner chain.) It's more common to take that situation and simplify it back to the circle-and-point system I described above. In fact, that's one of the steps used in a particular method of solving the Apollonius
Question: What is the center of the circle in the case of a hyperbola's directrix-like circle? Answer: The other focus.
Question: What is the ratio of distances in the case of a parabola's directrix? Answer: Always one.
Question: What is the name of the line that is perpendicular to the major axis of an ellipse? Answer: The Directrix Line. | 677.169 | 1 |
One of the criticisms of Euclid's parallel postulate was that it isn't simple.The statement of this proposition, I.30, is much simpler, and Playfair's axiom is much simpler. As they're each logically equivalent to Euclid's parallel postulate, if elegance were the primary goal, then Euclid would have chosen one of them in place of his postulate. Perhaps the reasons mentioned above explain why Euclid used Post.5 instead.
Question: Which of the following is simpler than Euclid's parallel postulate? (a) Playfair's axiom (b) Euclid's statement (c) Both are equally simple Answer: (a) Playfair's axiom | 677.169 | 1 |
Given two sides and the angle included by the two given sides, we can apply the Law of Cosines to find the missing side.
Spelling out the detail
Let's say the sides we know are and and the included angle is
We want to find the missing side
We know:
We have and . We can therefore work out the missing side as
Don't be confused by the different letters we are using here to the original statement of the law of cosines! Think of the law of cosines as being a relation between two sides and the included angle, to the remaining side. We have many choices as to how we label the sides and angles.
At this point we will know a side and an opposite angle. That's where the law of sines comes in. It says that in any triangle the ratio of the length of a side to the sine of the opposite angle is the same for all three sides. We can therefore use the law of sines to find the sine of the angle opposite each side that we know. Then, knowing the sine of the angle we can work out the angle itself.
Spelling out the detail
We know the ratio:
is the same for the other angles and sides. So in particular:
and
We know the quantities on the right. We can calculate the quantities on the left from these. Take the inverse sines and we're done.
Not quite.
We need to check that the angles and add up to 180o.
You might be horrified sometime to find that they don't. What can go wrong? See the next box.
Ambiguity of inverse sin
What is ?
What is ?
(you are expected to use a calculator for these).
Do you see the problem?
The inverse sine function 'makes the wrong choice' when doing the inverse sine of an obtuse angle. So, in solving an ASA triangle, you need to spot if one of the angles is obtuse. If the inverse sine for it tells you an acute angle you can then correct it to the equivalent obtuse angle.
The angles with the same sine add up to 180, so you can correct the angle that came out acute that should be obtuse by subtracting it from 180o. Alternatively, since at most one of the angles in a triangle can be obtuse, the one opposite the longest side, calculate the other angle instead. Then use the fact that the angles sum to 180o to get the remaining one.
Question: What is the included angle between these two known sides? Answer: The included angle is | 677.169 | 1 |
The dominions of a certain Eastern monarch formed
a perfectly
square
tract of country.
It happened that the king one day discovered that his
four sons were not only plotting against each other, but were in secret
rebellion against himself. After consulting with his advisers he
decided
not to exile the princes, but to confine them to the four corners of
the
country, where each should be given a triangular territory of equal
area,
beyond the boundaries of which they would pass at the cost of their
lives.
Now, the royal surveyor found himself confronted
by great
natural
difficulties, owing to the wild character of the country.
The result
was
that while each was given exactly the same area, the four triangular
districts were all of different shapes, somewhat in the manner shown in
the illustration.
The puzzle is to give the three measurements for
each
of the four districts in the smallest possible numbers—all
whole
furlongs.
In other words, it is required to find (in the smallest
possible numbers) four rational right-angled triangles of equal area.
Question: What was the requirement for the four triangular districts? Answer: They were to be four rational right-angled triangles of equal area. | 677.169 | 1 |
1 Answer
A few random thoughts (maybe they help you find a better solution) if you're using only the original sizes of the shapes:
as you point out, all shapes in the tangram can be made composed of e.g. the yellow or pink triangle (d-g-c), so try also thinking of a bottom-up approach such as first trying to place as many yellow triangles into your shape and then combine them into larger shapes if possible. In the worst case, you'll end up with a set of these smallest triangles.
any kind triangulation of non-polygons (such as the half-moon in your example) probably does not work very well...
It looks like you require that the shapes can only have a few discrete orientations. To find the best fit of these triangles into the given shape, I'd propose the following approximate solution: draw a grid of triangles (i.e. a square grid with diagonal lines) across the shape and take those triangles which are fully contained. This most likely will not give you the optimal coverage but then you could repeatedly shift the grid by a tenth of the grid size in horizontal and vertical direction and see whether you'll find something which covers a larger fraction of the original shape (or you could go in steps of 1/2 then 1/4 etc. of the original grid size in the spirit of a binary search).
If you allow any arbitrary scaling of the shapes you could approximate any (reasonably smooth ?) shape to arbitrary precision by adding smaller and smaller shapes. E.g. if you have a raster image, you can e.g. choose the size of the yellow triangle such that two of them make a pixel on the image and then you can represent any such raster image.
Question: Which two shapes can be composed of the smallest triangles in the tangram? Answer: The yellow and pink triangles (d-g-c)
Question: Can a half-moon shape be accurately triangulated using the tangram pieces? Answer: No | 677.169 | 1 |
Plane Geometry, The Pythagorean Theorem
The Pythagorean Theorem
If the legs of a right triangle have lengths a and b,
and the hypotenuse has length c,
the lengths satisfy a2 + b2 = c2.
For example, set a = 33 and b = 56,
and the hypotenuse has length 65.
This is the pythagorean theorem.
If the lengths of a right triangle are rational,
multiply through by the common denominator
and the lengths become integers.
Three integers that satisfy a2 + b2 = c2 are called a pythagorean triple.
All such
triples
have been characterized.
The simplest triple is a=3 b=4 c=5, also known as the 3 4 5 triangle.
Another example is 12 5 13.
Given a right triangle with sides a b and c,
draw a square a+b units on a side.
Place a copy of the right triangle in each of the four corners of the square.
Each triangle points to the next one, like a snake chasing its tail.
Now the bottom of the square, a+b in length,
is covered by the a leg of one triangle and the b leg of the next,
and similarly for the other three sides.
The region enclosed by the four triangles is a square, c units on a side.
This inner square is tilted relative to the outer square,
but it is still a square,
having area c2.
The four triangles have a combined area of 2ab,
and the outer square has area (a+b)2.
Put this all together and derive a2 + b2 = c2.
Question: What is a Pythagorean triple? Answer: Three integers that satisfy a² + b² = c² are called a Pythagorean triple. | 677.169 | 1 |
seems easiest to me to notice the relationship between the statements and triangles. The sum of two sides of a triangle is always greater than the third side, but the sum of the squares of the sides is not always greater than the square of the third side- the Pythagorean Theorem tells us that the sum of the squares of the two shortest sides is never greater than the square of the longest side if we look at the sides of a right angled triangle. That is, we've all seen dozens of examples (right angled triangles) where a+b > c but a^2 + b^2 = c^2.
So, let x^2, y^2 and z^2 be the sides of a 3-4-5 triangle and you get an example immediately that proves the answer to the original question should be E:
Question: How many examples of right-angled triangles have we seen where a+b > c but a^2 + b^2 = c^2? Answer: Dozens | 677.169 | 1 |
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The Office of Square Trading, the government body overseeing the sale of all rectangular shapes, has been investigating the illegal sale of squares.
"There is a massive black-market trade in squares, oblongs and rectangles," said Rex Tangle, chair of the Office of Square Trading. "People are circumventing the traditional markets of squares, and instead buying them under the counter."
This illegal trading is costing the government millions in lost Shape Tax revenue.
"Our sister organisation, the Office of Elliptical Trading, has not seen this same rise," said Tangle. "Where we're seeing people secretly buying rectangular shapes without paying the necessary shape tax, circles, ovals and ellipses are still being sold legitimately."
This tax evasion scam has gone largely unnoticed for so long due to a mislabelling of shapes, according to Tangle.
"We've seen perfect squares being sold is polygons," he said. "All polygons are tax exempt, due to them not being luxury shapes. We even saw one square being sold as two triangles to avoid paying tax. I think this is why circles and ovals have avoided being scammed this way, as it takes an infinite number of triangles to make a circle. And that would be expensive, even if the triangles are sold at a penny per triangle."
New procedures are being put into place to crack down on this illegal trading in squares, forcing the clear labelling of squares as squares and not as multiple triangles or a four sided polygon. Additionally, the Office of Square Trading is to be given new powers to seize and detain anybody falsifying the shape being sold
Question: What is one method used to mislabel squares? Answer: Selling them as polygons | 677.169 | 1 |
g. The question asked is: Is it the case that the side of any square multiplied by itself does equal half the diagonal line of the same square extended from any corner to its opposite corner when it is multiplied by itself. Since that would seem to be the same situation as that described in point c. above.
h. To clarify: 1.Given a line of six units the square is thirty six. And half the line is three units the square of such being nine, which when doubled is eighteen. And eighteen is exactly half of thirty six. 2. Given one side of a square. And given two diagonal lines drawn from two adjacent corners of the side to the centre of the side. And given that the square of one diagonal is added to the square of the other diagonal. Then the combined amount of the square of the two diagonals does equal the amount of the square of one side. 3. And either one of the diagonals could be extended through the centre of the side to the opposite corner. 4. And that diagonal line that extends from one corner of the side to the opposite corner of the side when squared must equal double the amount given in point 2.
Therefore if the previous hypothesis is correct it suggests that there is a constant relationship between the square of the side of any square and the square of the diagonal between two corners of the same square, in a proportional relationship.
{{{ This note contained within three brackets is a later addition to the original article. If the circumference of the circle is twenty two units, where a unit could be any constant length. And if in rough terms the diameter of the circle is seven units. Then the diagonal of the square would be seven units length. Since that is the hypotenuse of a triangle formed from one half of the square that would mean that one side of the square is five units length, in rough terms. Because, seven squared is forty nine. And five squared is twenty five. So by adding the square of two sides opposite the hypotenuse together we obtain fifty which is only one unit off forty nine. Furthermore if we were to fold one side of the square by ninety degrees to obtain a single line of ten units length then the square of that single line would be one hundred which is twice the square of half the line plus the square of half the line. Given that it means that there is a rough figure of five which must be relative to the seven and twenty two of pi. Because if the circumference is twenty two, the diameter is seven, and if the diameter is seven then the side of the square is five. So twenty two divided by five is four point four. Then a relationship between 4.4 : 3.14 must exist.
Question: What is the square of a side of a square given as 36 units? Answer: 12 units (half of 36)
Question: Is the statement "The side of any square multiplied by itself does equal half the diagonal line of the same square extended from any corner to its opposite corner when it is multiplied by itself" true? Answer: Yes, as proven in point h.
Question: If the circumference of a circle is 22 units, what is the approximate length of the diagonal of the square whose side is the diameter of the circle? Answer: 7 units | 677.169 | 1 |
How thick and long is the dowel? Given the right proportions Rob may be a winner.
Note the triangle is not connected to the square as end needs to be connected to end but the triangle does not need to be connected to the square?
Note while if we assume infinite thinness of the dowel the triangle has to be larger than the square for this to work if the dowel has the right thickness this can be accounted for.
Question: What is the thickness and length of the dowel mentioned in the text? Answer: The text does not provide specific measurements for the thickness and length of the dowel. | 677.169 | 1 |
A student seeks help coding a spherical navigation spreadsheet program. Doctor Vogler helps him
develop an algorithm that accounts for the trigonometry involved, with each drawing
on archived conversations.
I am researching Tycho Brahe and have come upon an example where he
uses [sin(a + b) + sin(a - b)]/2 to multiply large numbers together
due to the availability of sine tables. Can you explain the method?
A student wants to know how to unambiguously interpret strings of trigonometric
functions, multiplication, and exponentiation. Doctor Peterson digs into a history book
-- as well as another math doctor's conversation -- to illuminate the vagaries of the
notation.
A particle is projected with initial velocity v and angle theta in a
parabolic path. How can I show that at time t, when the angle to the
horizontal is gamma, tan(gamma) = tan(theta) - (gt/v) sec(theta)?
Question: What is the force acting on the particle that causes it to follow a parabolic path? Answer: The text does not provide this information, but it is likely gravity (g) | 677.169 | 1 |
Geometry (Encyclopedia of Science)
The term geometry is derived from the Greek word geometria, meaning "to measure the Earth." In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. Most people today know what those features are: lines, circles, angles, triangles, squares, trapezoids, spheres, cones, cylinders, and the like.
Humans have probably used concepts from geometry as long as civilization has existed. But the subject did not become a real science until about the sixth century B.C. At that point, Greek philosophers began to express the principles of geometry in formal terms. The one person whose name is most closely associated with the development of geometry is Euclid (c. 32570 B.C.), who wrote a book called Elements. This work was the standard textbook in the field for more than 2,000 years, and the basic ideas of geometry are still referred to as Euclidean geometry.
Elements of geometry
Statements. Statements in geometry take one of two forms: axioms and propositions. An axiom is a statement that mathematicians accept as being true without demanding proof. An axiom is also called a postulate. Actually, mathematicians prefer not to accept any statement without proof. But one has to start somewhere, and Euclid began by listing certain statements as axioms because they...
(The entire section is 1652 words.)
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Question: Who is the most famous person associated with the development of geometry? Answer: Euclid
Question: What is the origin of the term 'geometry'? Answer: The term 'geometry' is derived from the Greek word 'geometria', meaning "to measure the Earth". | 677.169 | 1 |
Dean Culver explains two methods for finding the center of a triangle in AutoCAD.
"I have found that there is usually more than one way to accomplish a task in AutoCAD. For example, I occasionally need to find the center or centroid of a triangle, and I've found two different ways to do so.
"First, I will use AutoCAD's internal calculator to place a point at the center of the triangle. I like to place a point there so I have something I can reference later if I need to dimension to it. Set the PDMode to the desired configuration (I prefer the setting of 3; that way, X marks the spot). Start by entering Point at the Command line, followed by the Enter key. It will then ask you to specify a point. Type in 'CAL followed by the Enter key, which will activate AutoCAD's internal calculator (the apostrophe before the command allows it to be operated transparently while you are in another command). Next, AutoCAD will ask you for the >>>> expression: Type in the expression (INT+INT+INT)/3 followed by the Enter key. Your mouse pointer will be replaced with a pick box; carefully select the three intersections of the triangle. Although it may not appear to be picking anything, rest assured it is picking the intersections, provided your aim is good and they do indeed intersect. It will place a point at the center or centroid of the triangle.
"The second method to find the center of a triangle is to turn the triangle into a region. Start by entering Region at the Command line, followed by the Enter key. It will then ask you to select objects: pick the three sides of the triangle, if the triangle was made using lines, or the whole triangle if it is a pline, then use the Enter key to finish the command. Next use the MassProp command and select the newly created region. AutoCAD will list the x and y coordinates of the centroid of the triangle in the AutoCAD Text Window." Notes from Cadalyst Tip Reviewer Brian Benton: This tip actually has five tips inside it: The first is that there is usually more than one way to do something in AutoCAD. The second and third are the tips on finding the center of a triangle. (A third method to find the center is to draw a line from one vertex to the midpoint of the opposite leg. Do this again on another vertex/leg. Where your two lines intersect will be the center. But that's a geometry tip, not a CAD tip.) A fourth tip is to use the Cal command inside an active command. And yet another tip is to use the MassProp command to find information about objects (regions, solids, etc.).
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User comments
Comment by Cooper,Kent
Posted on 2012-02-27 08:44:43
Just be sure you know WHICH center of the triangle you need. The 'CAL
method and the Region-centroid method will find the same location as the
Reviewer's described intersections of the medians. That's the centroid,
and the "center of gravity" of the triangle. But there are other "centers"
Question: What command is used to turn the triangle into a region? Answer: Region | 677.169 | 1 |
math find the coordinates of the missing endpoint given that p is the midpoint of NQ N(5,4) P(6,3) Q(3,9) P(-1,5)
algebra every inch on a model is equivalent to 3.5 feet on the real boat. what would be the mathmatical rule to express the relationship between the length of the model, m, and the length of the boat, b?
Geometry Which equation can be used to find the unknow ange measures? The diagram is a triganle. A= x+30, B= 3x, and C = x-15 A 5x+15=90 B 5x+45-180 C 3x^3+15=180 D 5x+15=180 E 3x+30=180
Question: If the model is 12 inches long, what would be the length of the real boat in feet? Answer: 42 feet | 677.169 | 1 |
thats correct. In such questions very important to note whether points are collinear or not.
Praet, try to solve this and u will know when to subtract sides:
How many triangles can be formed by connecting vertices of a hexagon such that no side of triangle coincides with that of hexagon.Hope this helps.
absolutely agree. the paints are not collinear and the question does not asks to count different triangles, so its solution should be simple 3C6=20.
Chose a point and then chose 2 out of 5 remaining points
so we have 5C2 per point. For 6 points we have
6 * 5C2 triangles.
Since triangle ABC is same as BCA and CAB we need to divide the combinations by 3
so total triangles = 6 * 5C2 /3 = 20
Question: How many ways can you choose 2 points out of 5 remaining points? Answer: 5C2 = 10 ways | 677.169 | 1 |
January 23rd 2009, 01:56 PM
Mush
Quote:
Originally Posted by augmata
Hello, everyone!Let denote the rotational speed of wheel 1.
Let denote the rotational speed of wheel 2.
Let denote the rotational speed of wheel 3.
Hence!
Now:
Hence:
Hence:
If they ever meet up again, then
This implies that
It also implies that:
How can a wheel pass through an angle, half that angle, and one ' ' of that angle simultaneously?
January 24th 2009, 10:20 PM
badgerigar
Quote:It is true.
The first 2 wheels will line up whenever the first wheel reaches the starting point and not otherwise. So the first 2 wheels will line up only every time the first wheel has undergone a whole rotation. in this time, the third wheel will have undergone rotations. Adding an integer number of rotations will never produce a whole number because is irrational. Therefore the 3 points will never line up again.
Question: According to the first quote, what happens when the three wheels meet up again? Answer: They cannot meet up again simultaneously. | 677.169 | 1 |
In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found; for example, using 120 cm it is possible to form exactly three different integer sided right angle triangles.
120 cm: (30,40,50), (20,48,52), (24,45,51)
Given that L is the length of the wire, for how many values of L ≤ 1,500,000 can exactly one integer sided right angle triangle be formed?
Project Euler changes the limits from time-to-time to make the problem more challenging or to discourage 'cheating'. They do publish a notice at the bottom of each affected problem description to warn of changes.
"Note: This problem has been changed recently, please check that you are using the right parameters."
Question: What is the purpose of the Project Euler problem described in the text? Answer: To find out for how many values of L (where L is the length of the wire) exactly one integer sided right angle triangle can be formed. | 677.169 | 1 |
algebra 2
300 pts pendingThis question
has expired. 24 hours after the asker selects a Best Answer, the points will be awarded.
thiasking145
What is the intersection of a cone and a plane parallel to a line along the side of the cone? Provide mathematical examples to support your opinions. You may use
equations, diagrams, or graphs to organize and present your thoughts.
Question: What would the intersection look like if the plane were to intersect the cone's tip? Answer: It would be a single point | 677.169 | 1 |
2nd statement says us that the sum of squares of 2 sides (side AB and AC) is greater than the third side. This statement implies only and only that the angle contained by AB and AC is acute, i.e angle BAC is acute. Since this angle is acute, one of the other 2 angles can be greater than 90 (as in 30-30-120) or less than 90 (60-60-60) or equal to 90 (30-60-90). Thus this statement is insufficient as we are not sure whether the other angle is greater than or less than 90.
Ok so in statement 1; a unique ratio tells us that shape is fixed and hence angles are fixed but can we actually determine if it's an acute or an obtuse triangle; I mean do we need to determine the actual angles from the given ratio to prove sufficiency.
Also, using the ratio of the sides as given in statement 1, how can we determine the actual angles?Each statement alone is sufficient to answer the question...
Regards, Jack
Regarding statement 1, "the angles of a triangle are proportional to the length of the sides."
This is definitely not true! There is no such theorem.
The so-called "sinus theorem" states that for any triangle, the following equality holds:
\frac{AB}{sinC}=\frac{BC}{sinA}=\frac{AC}{sinB}=2R, where A, B, Cdenote the angles of the triangle and R is the radius of the circumscribed circle. sin is the sinus function, and the value of sina is not proportional to the value of a.
From statement (1) we can deduce that AB = 6x, BC = 4x, AC = 3x, for certain positive x. Then AB^2=36x^2>BC^2+AC^2=25x^2, which means the triangle is obtuse angled, with angle C greater than 90. Therefore (1) is sufficient.
(2) is not sufficient, as it only proves that angle A is not obtuse, but nothing is known about the other two angles.
Question: Is the statement in the first paragraph about the sum of squares of two sides of a triangle being greater than the third side always true? Answer: No, this statement is only true for obtuse triangles.
Question: What can we deduce about the triangle from the information in the first statement (AB = 6x, BC = 4x, AC = 3x)? Answer: We can deduce that the triangle is obtuse-angled, with angle C greater than 90 degrees.
Question: Is the statement in the first paragraph of the text about the angles of a triangle being proportional to the length of the sides true? Answer: No, this statement is not true. | 677.169 | 1 |
The figure shows a tangential
quadrilateral ABCD. Diagonals AC and BD meet at E. If r1,
r2, r3, and r4, are the radii
of the incircles of triangles ABE, BCE, CDE, and ADE, prove that
.
Geometry problem solving
Geometry problem solving is one of the most challenging skills for
students to learn. When a problem requires auxiliary construction, the
difficulty of the problem increases drastically, perhaps because
deciding which construction to make is an ill-structured problem. By
"construction," we mean adding geometric figures (points, lines, planes)
to a problem figure that wasn't mentioned as "given."
Question: Is the quadrilateral ABCD a tangential quadrilateral? Answer: Yes | 677.169 | 1 |
Position and velocity vector of point referred to equinox find figure 4 a bit puzzling. The text uses lower case phi, a vertical line with a circle in the middle, but the figure shows what is perhaps a capital phi. A looping curve. I'm guessing this is meant to be former symbol, vertical + circle,
Secondly, it looks like the x,y,z axes for the topo system should be labeled with a prime on x,y,z.
How to get the Cartesian coordinates (x,y,z) of the point in terms of its spherical coordinates (r,θ,ϕ)?
It's clear from the geometry that the z-coordinate of the point is just given by z = rcosθ. (Just take the position vector of the point and project it onto the z-axis). EDIT: or just look at the right triangle formed by the dashed lines.
The magnitude of the projection of the position vector onto the xy-plane is just rsinθ. It's the other side of the (dashed) right triangle that we used above to get the component of the vector that was parallel to the z-axis. So it's the dashed line on the xy-plane in the figure.
Now, take this *projected* vector and further resolve IT into an x-component and a y-component on the plane. The x-component will be given by the projected vector multiplied by the cosine of the angle between that vector and the x-axis:
x = (rsinθ)cosϕ
Similarly:
y = (rsinθ)sinϕ
Again, you get these by drawing the appropriate right triangle on the plane (not shown).
EDIT: The meanings of theta and phi are SWITCHED in my figure compared to yours. Sorry for any confusion. Mathematicians and physicists often use the opposite symbols for these two angles. gave you the derivation for 12 in my previous post. Equation 13 is just obtained by differentiating equation 12 with respect to time, and using the chain rule. The equation makes perfect sense if you assume that [itex]\phi_A^\prime [/itex] is constant i.e. it does not vary with time. This makes sense since it is supposed to be the latitude of the station, which is fixed.
Equation 14 is just the same as equation 12, except in the station's coordinate system. Also, every component has been divided by the total length of the vector ("R") since this is meant to be a UNIT vector in the direction of the zenith. That's why 14 looks like 12, but without the radius as a coefficient on each term.
Quote by solarblast
Secondly, it looks like the x,y,z axes for the topo system should be labeled with a prime on x,y,z.
I'm sorry, I don't know what you mean here.
Quote by solarblast
Thirdly, the two phi's look like they should have the primes swapped.
Question: How is the y-coordinate of a point calculated from its spherical coordinates (r,θ,ϕ)? Answer: y = (rsinθ)sinϕ
Question: How is the x-coordinate of a point calculated from its spherical coordinates (r,θ,ϕ)? Answer: x = (rsinθ)cosϕ
Question: What is the magnitude of the projection of the position vector onto the xy-plane? Answer: rsinθ
Question: What is the purpose of equation 14 in the text? Answer: To represent a unit vector in the direction of the zenith in the station's coordinate system | 677.169 | 1 |
Stair Parts Fig. 26 shows the development of the angle of tangents for the face mould and the bevel for springing the wreath. Draw the angle ABC on center line of rail as shown; draw dotted line from center O to B; draw DE at right angles to OB; from center B swing around A and C to D and E; set up one riser from D, to F, and one down from E; mark one step above and below the pitch-board; draw pitch-line XX.
Connect CA, and continue OB to G; with B as a center, describe an arc touching CA; from G as a center with the same radius, describe an arc; from E draw line touching this arc; from G again swing around GF to H; connect GH, and d the angle of tangent is complete. The amount of straight wood on wreath is shown from E and H to the joints XX.
Stair Parts Fig. 27
To obtain the bevel it is first necessary to find the point I; from H with FD for radius, swing an arc and intersect with another from E; having CA for radius, connect El and HI; take a center, K, anywhere on the line EH, draw an arc touching GH and cutting EH at L; square down from K, cutting JH at M; connect LM, and the bevel is found for both joints of the wreath, the pitch being one straight line. Stair Parts Fig. 27 shows application of bevel to wreath.
Stair Parts Fig. 28
Stair Parts Fig. 29-30
Question: What is the purpose of the figures 29 and 30? Answer: Figs. 29 and 30 are not provided in the text. | 677.169 | 1 |
Triangles can be classified in two ways: by their sides and by their angles. Classifying by sides is a bit easier, so let's start with that. There are three possibilities for a triangle when dealing with their sides.
A summary of Classifying Triangles in 's Special Triangles. Learn exactly what happened in this chapter, scene, or section of Special Triangles and what it means. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans.
Math: Geometry: Angles and Triangles ... Classify Triangles. Try to identify these different types of triangles. translate link. ... translate link. Figure It Out! This table game can help kids to learn how to identify different types of triangles. translate link. Impossible Triangles.
Tool: Triangle Classification Game Screenshot: Go to: (opens a new window) Description: In this activity the interactivity attempts to train the user in the classification of triangles with respect to sides and internal angles
Question: How many possibilities are there when classifying triangles by their sides? Answer: Three. | 677.169 | 1 |
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