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Unformatted text preview: Math 131 Final Exam Instructions: You are to take these problems home. You must do every problem. They are due at
10:00, December 16th, 2002. You may use as resources ONLY your own copy of the textbook, your own
class notes, ﬁles from the course homepage and the Geometers Sketchpad program (but only ﬁles you
create yourself). You may not discuss ANY aspect of this exam with anyone other than the instructor. Turn in this problem sheet (signed on page 2) along with your answer to the problems. The sheets
should be stapled or clipped together. You do not have to type your work, but it should be easy to read. 1. (10 points) A student has the following method for bisecting an angle A C with compass and straightedge. Decide the the method is valid and explain why it does or doesn’t
work: ' Pick a point on BA and call it D; pick one on BC and call it E. Open your compass
to the length of DE. Place the point of your compass on D and draw a circle. Now place
the point of the compass on E and draw a circle. Draw a line through the intersections
of these circles; call this line 1. Find the intersection of 1 and DE; call this F. Then BF is your angle bisector. 2. Given two congruent triangles with the same orientation, explain
(a) (5 points) how to determine (using only straightedge and compass) what type of rigid motion
sent one triangle to the other (b) (5 points) if it is a rotation, how to ﬁnd the center of the rotation using only straightedge and
compass 3. (7 points) Each vertex of an icosahedron is colored either Red, Green, Blue or Yellow. Decide if
the following statement is true, and justify your answer: There is an edge of the icosahedron that has endpoints (vertices) that are the same color. 4. (10 points) Describe all symmetries of pyramids which have bases that are polygons with an even
number of sides. . You have a square cake that is covered with frosting on the top and on all four sides, as shown in
this picture (a) (10 points) You want to share the cake between three people so that everyone gets equal
amounts of cake AND frosting. How could you cut the cake to do this? (b) (3 points) How could you cut the cake so that you either made the fewest number of cuts or
cut it into the fewest number of pieces (while still sharing equally between the three people)? . You are making an equilateral triangle kaleidescope. It consists of three congruent rectangular
minors with their edges touching, arranged to form an equilateral triangle (with the mirrored sides facing each other). (a) (5 points) If you want to ﬁt the mirrors in a tube (cylinder) that is 10 inches long and 2 inches
in diameter, what should the dimensions of the mirrors be? (b) (5 points) What 3D shape do the mirrors make? What are all the symmetries of this shape?
(c) (10 points) If the kaleidescope is placed over a drawing so it looks like this: what do you see in the kaleidescope? Describe this completely. I hereby aﬂirm that in solving the attached problems I have not used
any external resources other than my own copy of the textbook, my own class notes and papers, and the class website. I have not spoken
or written about these problems to anyone, whether a student in this class or not, even to the extent of indicating whether I solved a
problem, or how diﬂicult any or all of it was. (signature) Math 131 Final Exam Instructions: You are to take these problems home. You must do every problem. They are due at 10:00, December 16th, 2002. You may use as resources ONLY your own copy of the textbook, your own
class notes, ﬁles from the course homepage and the Geometers Sketchpad program (but only ﬁles you
create yourself). You may not discuss ANY aspect of this exam with anyone other than the instructor.
Turn in this problem sheet (signed on page 2) along with your answer to the problems. The sheets
should be stapled or clipped together. You do not have to type your work, but it should be easy to read. 1. (10 points) A student has the following method for bisecting an angle A C with compass and straightedge. Decide the the method is valid and explain why it does or doesn’t
work: ‘ Open up your compass to any distance and, with the point on B make a mark along BC; call this D. Then change the opening of your compass, place the point on D and make a circle. Without changing your compass, put the point on B and make a mark along BA; call this E. Now open your compass to length BD. With the compass point on E, make a circle. The circles intersect at a point inside the angle; the line that passes
through this intersection and point B is the angle bisector. 2. Given two congruent triangles with the same orientation, explain (a) (5 points) how to determine (using only strai ghtedge and compass) what type of rigid motion
sent one triangle to the other 3. (7 points) Each vertex of an icosahedron is colored either Red, Green, Blue or Yellow. Decide if
the following statement is true, and justify your answer: There is an edge of the icosahedron that has endpoints (vertices) that are the same color. 4. (10 points) Describe all symmetries of pyramids which have bases that are polygons with an odd
number of sides. 5. You have a square cake that is covered with frosting on the top and on all four sides, as shown in
this picture (a) (10 points) You want to share the cake between three people so that everyone gets equal
amounts of cake AND frosting. How could you cut the cake to do this? (b) (3 points) How could you cut the cake so that you either made the fewest number of cuts or
cut it into the fewest number of pieces (while still sharing equally between the three people)? 6. You are making an equilateral triangle kaleidescope. It consists of three congruent rectangular
minors with their edges touching, arranged to form an equilateral triangle (with the mirrored sides facing each other). (a) (5 points) If you want to ﬁt the minors in a tube (cylinder) that is 10 inches long and 2 inches
in diameter, what should the dimensions of the mirrors be? (b) (5 points) What 3D shape do the mirrors make? What are all the symmetries of this shape?
(c) (10 points) If the kaleidescope is placed over a drawing so it looks like this: what do you see in the kaleidescope? Describe this completely. I hereby aﬁirm that in solving the attached problems I have not used
any external resources other than my own copy of the textbook, my
own class notes and papers, and the class website. I have not spoken
or written about these problems to anyone, whether a student in this
class or not, even to the extent of indicating whether I solved a problem, or how diﬁcult any or all of it was. (signature) Math 131 Final Exam Instructions: You are to take these problems home. You must do every problem. They are due at
10:00, December 16th, 2002. You may use as resources ONLY your own copy of the textbook, your own
class notes, ﬁles from the course homepage and the Geometers Sketchpad program (but only ﬁles you
create yourself). You may not discuss ANY aspect of this exam with anyone other than the instructor. Turn in this problem sheet (signed on page 2) along with your answer to the problems. The sheets
should be stapled or clipped together. You do not have to type your work, but it should be easy to read. 1. (10 points) A student has the following method for bisecting an angle A C with compass and straightedge. Decide the the method is valid and explain why it does or doesn't
work: Open your compass to any length and, with the point on B make a mark along BC;
call this D. Then put the point on D and make a circle. Now change the opening of your
compass. Put the point on B and make a mark along BA; call this E. Put the point of
your compass on E and make a circle. The circles intersect at a point inside the angle;
the line that passes through this intersection and point B is the angle bisector. 2. Given two congruent triangles with the same orientation, explain
(a) (5 points) how to determine (using only straightedge and compass) what type of rigid motion
sent one triangle to the other
(b) (5 points) if it is a rotation, how to ﬁnd the center of the rotation using only straightedge and
compass
3. (7 points) Each vertex of an icosahedron is colored either Red, Green, Blue or Yellow. Decide if
the following statement is true, and justify your answer:
There is an edge of the icosahedron that has endpoints (vertices) that are the same color. 4. (10 points) Describe all symmetries of prisms which have bases that are polygons with an even
number of sides. 5. You have a square cake that is covered with frosting on the top and on all four sides, as shown in
this picture I (a) (10 points) You want to share the cake between three people so that everyone gets equal
amounts of cake AND frosting. How could you cut the cake to do this? (b) (3 points) How could you cut the cake so that you either made the fewest number of cuts or
cut it into the fewest number of pieces (while still sharing equally between the three people)? 6. You are making an equilateral triangle kaleidescope. It consists of three congruent rectangular
mirrors with their edges touching, arranged to form an equilateral triangle (with the mirrored sides facing each other). (a) (5 points) If you want to ﬁt the mirrors in a tube (cylinder) that is 10 inches long and 2 inches
in diameter, what should the dimensions of the mirrors be? (b) (5 points) What 3D shape do the mirrors make? What are all the symmetries of this shape?
(c) (10 points) If the kaleidescope is placed over a drawing so it looks like this: what do you see in the kaleidescope? Describe this completely. I hereby aﬁrm that in solving the attached problems I have not used
any external resources other than my own copy of the textbook; my
own class notes and papers, and the class website. I have not spoken
or written about these problems to anyone, whether a student in this
class or not, even to the extent of indicating whether I solved a
problem, or how difﬁcult any or all of it was. (signature) ...
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This note was uploaded on 08/08/2008 for the course MATH 131 taught by Professor Allprofs during the Fall '07 term at University of Wisconsin.
 Fall '07
 ALLPROFS

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