### MATH325: Sample Test 3

Name:

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 hyperbolic geometry, the circumference of a circle of radius r is

a) greater than 2πr.
b) less than 2πr.
c) undefined.
d) a function of the location of the center of the circle.

2. (25 pts) List the steps needed to construct an angle bisector in hyperbolic geometry or explain why the method used in the Euclidean plane does not work in the hyperbolic plane.

3. (25 pts) Describe two ways in which the axioms of spherical geometry differ from those of Euclidean geometry.

4. Imagine doing the following in hyperbolic geometry:
• Given points A and B in the hyperbolic plane,
• Construct CA(|AB|) and CB(|AB|).
• Let points M and N be the two points of intersection of the two circles.
• Construct line MN.

a) (10 pts) Prove or explain why you cannot: every point P on line MN has the property |PA| = |PB|.

b) (10 pts) Prove or explain why you cannot: every point P that has the property |PA| = |PB| is on line MN.

5. In taxicab geometry, define Er(AB) to be the set of all points P such that |AP| + |BP| = r.

a) (10 pts) Draw E6(AB) for two points A and B that are three blocks apart in the x-direction and one block apart in the y-direction.

b) (10 pts) Read the definition of an ellipse posted at http://www.mathopenref.com/ellipse.html. Would you say that Er(AB) is an ellipse in taxicab geometry? Write a paragraph supporting your answer.

This practice test should be similar to but harder than the actual test to be given in class. Also, the actual test will include space for you to work problems in.

Sample solutions (please notify me if you find errors).

Outcomes:

• Know Euclid's five axioms and the implications of accepting, changing or omitting an axiom.
• Prove theorems based on a given set of axioms, lemmas, and previously proved theorems.
• Be able to follow or provide step by step instructions for geometric constructions.