Problems for Week 8

Quiz on Friday!

Read Brown, Transformational Geometry, Chapter 1.

Exercises 13 (Due Wed, 11/17)

Most of these will be done in class, Monday, so you only need to write them up.

E13.1 Brown, p. 7 #1

E13.2 Brown, p. 7 #2

E13.3 Brown, pp. 15-16 #14

E13.4 Brown, pp. 15-16 #15

E13.5 Brown, pp. 15-16 #11

E13.6 Brown, pp. 15-16 #12

E13.7 Brown, pp. 15-16 #13

Exercises 14 (Due Friday, 11/19)

Exercises 14 (Due Friday, 11/19)

E14.1 Brown, pp. 20-22 #1

E14.2 Brown, pp. 20-22 #2

E14.3 Brown, pp. 20-22 #4

E14.4 Brown, pp. 20-22 #6 (this will be done in lab Wed)

E14.5 Brown, pp. 20-22 #7 (this will be done in lab Wed)

Assignment 8 (Due Monday, 11/22, 70 points)

8.1 A pool table problem (10 points)

Brown, p. 21 #5. Discuss carefully. Give reasons.

8.2 Constructing a square using isometries (15 points)

Brown, p. 22 #10. Carry out the construction in an example. Explain how and why.

8.3 Polar coordinates and line reflection (10 points)

Given a point O on a plane, explain how to define a polar coordinate system with O as the center. Given a point U at distance R from O with polar angle q . Let m be a line through O with polar angle a . If U’ is the reflection of U in m, write down the polar coordinates of U’

8.4 Double and triple line reflections in polar coordinates (20 points)

Use what you developed in 8.3. Let a, b, and c be three lines through O, having polar coordinate angles of a, b and g at O. Let U be a point at distance R from O with polar angle q. If reflection in a, b, and c is denoted by A, B, and C, respectively, then give formulas in polar coordinates for the following.

  1. The double reflection BA(U). What kind of isometry is BA and how can this be seen from the formula.
  2. The triple reflection CBA(U). What kind of isometry is BA and how can this be seen from the formula.
  3. The quadruple reflection (AB)(BA)(U). This should come out simple. Why?
  4. Based on (a) and what you know about isometries, without computing another formula, what kind of isometry would DCBA be, if D is reflection in another line through O. How about a composition of 5 isometries (use (b) in your reasoning)?

8.5 Rotations and equilateral triangles (15 points)

Do Brown, p. 30, #5 and #7. Construct a triangle ABC as in #5 but put an equilateral triangle ACZ on side AC. What are the angles between line BZ and lines XC and AY. What do you observe about how these lines intersect?

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