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Unformatted text preview: Math 131 May 13, 2004
Final Exam Name: There are 150 possible points on this exam. Be sure to read each question
carefully and answer the question asked. Show your work neatly and clearly—
answers without justiﬁcation may reduce your score. Partial credit may be
given for a correct approach even if you don’t get to the right answer. You
may use calculators, compasses, and straightedges. No notes, textbooks, or
other materials are allowed. 1. A Greek Adventure
You wake up one morning to find yourself surrounded by ancient Greeks. ‘ (No, you are not at a class reunion on Langdon Street.) You are asked
for your passport, which you don’t have, and are promptly arrested.
At your trial, you are forced to demonstrate your knowledge of Greek
culture by doing the following two tasks. (Remember this is your o’nly
chance to save yourself from the wretched dungeon of geometric illiter— acy! (a) (20 points) State and prove t
ion is orderly and cl he Pythagorean Theorem. Make sure
your presentat ear to the skeptical observers. (b) (15 points) The prosecutor says “If the sides of a triangle T have
lengths a, b, and c satisfying a2 + b2 = 02, then T is a right triangle.”
Then he demands “What do you think about that?!” If he is telling the truth, give a proof. If it is possible that he is lying,
give a counterexample to his claim. 2. (15 points) Given the point P on the line E, construct a line perpen—
dicular to E and passing through P. Justify your construction. 3. (15 points) How many symmetries does a square prism have? How
many of these symmetries can be expressed Without using any reﬂec—
tions? (That is, how many symmetries are obtainable using just rota tions?) Explain how you know your answer is correct. 4. (12 points) What are the 4 types of rigid motions of the plane? (No justiﬁcation required.) 5. (12 points) The following describe rigid motions of the plane. Assume
that neither (a) nor (0) is the “do nothing” symmetry. After each one,
indicate which type(s) of rigid motion it could be. No justiﬁcation is
required. (Just write the appropriate type or types from your list in Problem 4.) (a) Composition of 3 rotations (b) Composition of 3 reflections (c) A rotation followed by a translation oints on a sphere, how many lines (in the 6. (10 points) Given 2 distinct p
through the two points? Sketch the sense of spherical geometry) pass
possibilities. here a regular polyhedron whose faces are octagons? If 7. (12 points) Is t
ts name. If not, give an argument, for such a polyhedron exists, give i
why such a polyhedron is impossible. 8. (12 points) Write a short essay on non—Euclidean geometry in the space
provided. Be sure to address the following issues: a What is nonEuclidean geometry?
0 What is its early history?
0 Why is it important? y regular) polyhedron P has 20009:2 9. (15 points) Suppose a (not necessaril
List the possible number or num— vertices, 2003: faces, and 398 edges.
bers of vertices for P. 10. (12 points) A cube C is inscribed in a sphere S of radius r. Find (exactly) the ratio
volume of S volume of C’ ' You may use the fact that the volume of a sphere of radius r is gm”. (Your answer should be a number.) 11. (10 points Extra Credit) Inscribe a circle in the given triangle below. Justify your construction. 10 ...
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