Math 360, Dr. Vaden-Goad, Spring 2004, practice test 3, name __________________
Think carefully, show all work
Fold test lengthwise with essay on the outside
1. Write a brief essay (to be specified)
2-6 will be basic recall and short answer. You are responsible for all the words used
between page 148 and page 243, even if the definition occurs elsewhere. Additionally,
you are responsible to be able to list and describe the basic transformations: translation,
rotation, reflection, dilation, inversion, glide-reflection. You are also responsible to
sketch the image of a given transformation applied to a given subset of the domain.
examples:
1. If the circle of radius 4 centered at (5,3) is reflected about the line y=18, completely
specify its image.
2. If the triangle with vertices at (0,0),(3,0) and (0,4) is rotated 30 degrees
counterclockwise about the point (5,5), give the coordinates of the new vertices.
3. define Saccheri Quadrilateral, Clairaut’s axiom, Lambert Quadrilateral, limiting
parallel ray, Beltrami-Klein model, Poincare model (disk or half-plane), pole of the chord
m, Minkowski metric,
4. Give one piece of personal information about (one of those people).
5. Briefly discuss the contribution of (one of those people ) to geometry.
7-10 will be simple proofs or connections among the various approaches to geometry.
Proofs will be of a form where you must first decide whether the statement is true or false
in the stated context and then prove that you are correct. If the statement is independent,
you may give models demonstrating independence without proving your statements about
the models. In the examples that follow, the things in parentheses are there to help you
understand how to solve the example. They won’t be on the test.
examples:
Prove or Disprove: In the Poincare disk model, The Euclidean axiom is satisfied.
(Disprove: We know E is equivalent to H and H is false.)
Prove or Disprove: In the Cartesian plane, inversion is an isomorphism.
(Disproof: it isn’t even defined on the whole plane: the center of the circle
has no image.)
Prove or Disprove: In the Cartesian plane with the origin removed, inversion
about the unit circle centered at the origin is one to one.
(Proof: If two points have the same image, say (r, theta) in polar
coordinates, then they both have the polar coordinates (1/r,theta). Thus they are the same
point.)
Connection: Given two before/after pairs for an unspecified dilation of the Cartesian
plane, describe the steps you would follow to find the center of the dilation.
Connection: Provide Cartesian equations for the reflection about the line y = 2x.