Math 444 Quiz 2

• Part A: Answer all 4 questions in Part A.
• Part B: Answer ONE of 3 questions 5A, 5B, or 5C. Notice 5A and 5B count more points than 5C, so one of them is a better choice if the answer is correct.

Part A

Problem 1 (20 points) Composition of functions

In each case, tell what kind of isometry the product could be. Write ALL possible kinds but do not include kinds that you can exclude with the given information. The answers should be taken from the list of names of the types of isometries of the plane.

a) If S is a glide reflection and T is a rotation, what kind of isometry is ST?

b) If U is a point reflection and V is a translation, what kind of isometry is UV?

c) If E and F are glide reflections, what kind of isometry is EF?

d) If A is a rotation and B is a line reflection, what kind of isometry is AB?

Problem 2 (10 points) Write the equations of two lines u and v so that R_vR_u maps the shape on the left to the shape on the right. (Notation: R_u means reflection in line u.)

Equation for u:

Equation for v:

Problem 3 (25 points) Image of an angle

a) Write the definition of isometry.

b) Let T be an isometry. Suppose A, B, C be non-collinear points, and A' = T(A), B' = T(B), C' = T(C). Prove that the angles ABC and A'B'C' are congruent.

Note: You can use the definition of isometry and facts about figures in the plane, but no other theorems about isometries without proving them.

Problem 4 (20 points) Constructing defining data of an isometry

The two triangles in the figure are congruent.

a) Tell what kind of isometry T will map triangle ABC to the other triangle.

b) Construct with straightedge and compass the geometrical defining data for this isometry T.

c) Make clear from labeling or comments what your method is and what the data are, but you do not have to justify your method.

Defining Data: _____________________________________________

Part B.

Answer ONE of Problems 5A (25 points) or 5B (25 points) or 5C (15 points). You can only count one. Cross out the others if you work on more than one

Problem 5A (25 points) Composition of rotations

Given the points A and B below; let S be rotation with center A by 60 degrees and let T be rotation with center B by 180 degrees.

a) Tell what is the isometry U = ST. Be as explicit as you can.

b) Construct the geometric data defining U.

Problem 5B (25 points) Let M₁, M₂, M₃ be line reflections in the lines m₁, m₂, m₃ below. Let N = M₁ M₂ M₃.

a) Tell what isometry is N. Be as explicit as you can.

b) Construct the geometric data defining N.

Problem 5C (15 points) Let M₁, M₂, M₃ be line reflections in the lines m₁, m₂, m₃ below. Let N = M₁ M₂ M₃.

a) Tell what isometry is N. Be as explicit as you can.

b) Construct the geometric data defining N.