### MATH325 Test 3Spring 2010

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You may use pencil, pens, protractor, ruler, compass and scissors on this test. You may not use a calculator, book or notes. Please take your time on each problem and show your work in the space provided. If you get stuck on a problem, move on to the next and return when you have time.

1. (10 pts) Circle the letter of the response that best completes the sentence.

In spherical geometry, the sum of the measures of the interior angles of a triangle is

a) less than 180°.
b) greater than 180°.
c) equal to 180° unless the triangle is degenerate.
d) less than 180° for triangles that cover less than half the surface area of the sphere and greater than 180° for triangles that cover more than half the surface of the sphere.

2. (30 pts) Give two examples of axioms of hyperbolic geometry that are the same as those of Euclidean geometry.

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3. (30 pts) List the steps needed to construct a perpendicular bisector in hyperbolic geometry or else explain why the method used in the Euclidean plane does not work in the hyperbolic plane.

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4. Given two points A and B in taxicab geometry, let L(AB) be the set of all taxicab points P such that
|AP| = |BP|.

a) (20 points) One choice of A and B is shown below. Circle the points that would belong to L(AB) in this example.

An equivalent question phrased as a story problem: Your job is at location A and your spouse's job is at location B. You are about to buy a home together. You've agreed that your new home will be the same taxicab distance from each of your workplaces. Circle all the points that are possible locations for your new home.

b) (10 points) A friend of yours suggests that you define L(AB) (the set of possible locations of your new home) to be the perpendicular bisector of segment AB. Is this a good idea? Write a paragraph supporting your answer. (Answers that reveal an understanding of why this might be a good idea will receive more credit.)

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Bonus: (5 pts) Referring to the last question, suppose that L(AB) is defined to be a line in taxicab geometry. Use the definition of L(AB) to describe how to construct a line through an arbitrary pair of taxicab points C and D. (For partial credit, construct a line through any pair of points C and D of your choosing.)