stugfinex.htm

Study Guide for the Final Exam
Math 330B, Spring, 2004

The final exam will be similar to the preceding exams, only a page or two longer. There will be some that just ask you to state a definition, some that ask you to provide the reason(s) for each step of an argument, some "circle the best answer" questions and, of course, some of the proofs we have done in class. The `circle the best answer' questions may be based on anything that we have discussed this quarter. There may also be a couple things that you haven't seen; these, however, will be closely related to the material we have worked on in class.

Definitions: Fixed point, fixed line, rotation, translation, pole, half-turn, glide reflection, and pencil. Know the definitions of sine, cosine and the vector definitions of translation and reflection.

Know how to prove the following theorems using the axioms of transformational geometry

If and are fixed points of the reflection $R_{a}$ , then the line determined by and $\ Q$ is fixed by $R_{a}$ .
Every line perpendicular to the line is fixed by the reflection $R_{l}$ .
Let and be lines. Then if and only if .
Let and be two distinct lines. Then is perpendicular to if and only if .
If is a pole of and is perpendicular to , then is on .
If is a pole of and is on a line , then is perpendicular to
(First theorem on three reflections). If the lines and intersect at a point , then there is a line passing through such that .
If $M=R_{l}R_{m}$ is a rotation about , then given any line passing through there is another line passing through such that $M=R_{v}R_{u}$ .
If $H_{P}$ is a half-turn, then $H_{P}=R_{v}R_{u}$ for any pair of perpendicular lines and that meet at .
Every line passing through is a fixed line under $H_{P}.$
Every product $H_{P}R_{v}R_{u}$ can be replaced by a product $R_{l}R_{m}$ .
In an elliptic plane, every motion is a rotation.
In a Euclidean plane, every proper motion is either a translation or a rotation; every improper motion is a glide reflection or a reflection.

Also be able to -

Answer questions similar to the ones that appeared on the second homework assignment (putt-putt golf and symmetries of a figure.)
Describe the fixed points and fixed lines of any motion in the Euclidean plane.
Show that if and are parallel lines, then there is a vector $\overrightarrow{v}$ such that
Use the matrix representations of a rotation (of $\theta$ degrees) centered at the origin and the reflection $R_{l}$ when is a line passing through the origin that makes an angle of $\theta$ degrees with the origin. Be able to derive the matrix representation of a rotation of $\theta$ degrees when the rotation is centered at the origin.
Prove the Ruler Placement Theorem. Know the definitions of function, one-to-one, and onto.
Produce a coordinate system for a given line $l\$ using the equation of a line and any of the distance functions $d_{1},$ $d_{2}$ or $d_{\infty }.$

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