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Unformatted text preview: Math 131 December 19, 2003
Final Exam ‘ 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 tape, scissors, clay, models of polyhedra, calculators, compasses, and
straightedges. No notes, textbooks, or other materials are allowed. 1a 1. A Hellenic Flashback T W0 months after your midterm Greek adventure you have a dream.
You are back in the courtroom being asked to demonstrate your knowl
edge of Greek culture again. Will it be a nightmare? Or a rousing success? Only YOU can decide . . . (a) (20 points) State and prove the Pythagorean Theorem. Make sure
your presentation is orderly and clear 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 = (:2, 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) Construct a. 15° angle in the space provided. Be sure to
fully explain why your construction works. ‘ 3. (15 points) Given the line segments below of lengths 1, a, and b, con
struct either 3 or ab. Clearly indicate which one you are constructing, and which line segment has the desired length. No justiﬁcation is re
quired, but leave all arcs and lines used in your construction. 4. (12 points) What are the 4 types of rigid motions of the plane? (No
justiﬁcation required.) (i)
h<u>______________
(iii)
(iv) 5. (12 points) The following describe rigid motions of the plane. Assume
that (c) is not the “do nothing” symmetry. After each one, indicate
which type(s) of rigid motion it could be. No justiﬁcation is required.
(Just write (i), (ii), (iii), and/ or (iv) to indicate the types you listed in
Problem 4.) l (a) T($,y) : (y, _‘T)
(b) Composition of 3 reﬂections (c) A rotation followed by a translation 6. (10 points) How many symmetries does a (regular) octahedron have?
How do you know? 7. (9 points) Is there a regular polyhedron whose faces are octagons? Why
or why not? 8. (20 points) What is the measure of the central angle of a regular 72—gon?
The vertex angle of a regular 72gon? Is a regular 72gon constructible? Why or why not? 9. (10 points) (a) What are the lines in cylindrical geometry?
1 (b) Is it true that there is always a unique line passing through any
two distinct points (in cylindrical geometry)? Why or why not? 10. (12 points) Draw identiﬁcation diagrams for each of the following sur—
faces. (No justiﬁcation is required.) (a) Mobius strip (b) Torus (2 Doughnut) (0) Pro j ective Plane 11. (10 points Extra Credit) What is the largest odd number n for which
the regular ngon is known to be constructible? (You are not required to simplify your expression for n.) ...
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