Math 444 Quiz 4

NAME _______________________________________________

Problem 1 (Short Answer)

Suppose that each of E, F, G, H, J and K are glide reflections.  Let U = EFGHJK. 

With this information it is not possible to say precisely what kind of isometry U is, but one can narrow down the possibilities.  Circle each name of a kind of isometry below when U could be an isometry of that type (for some choice of glide reflections).  Then in a sentence, give a reason for your answer.

(a) Identity                    (b) Line Reflection                    (c) Point Reflection

(d) Translation              (e) Rotation                              (f) Glide Reflection

Reason:

Problem 2 (Short Answer) This figure is a regular pentagon.

(a)    How many rotation symmetries does this figure have? __________

(b)   How many line reflection symmetries does this figure have? ___________

(c)    Let line m = line OE, For what line n is R_mR_n = O₁₄₄? _____________

(d)   What symmetry is R_OBR_OER_OC? __________________________ (Give precise name or description)

Problem 3 (Answer and Construction)

Given the points A, B, C in the figure,

(a)    Tell what transformation T = A₁₈₀ B₁₈₀ C₁₈₀ is.

Answer:  T = __________________________________________

(b)   Construct (with straightedge and compass) the defining data for this transformation.

Problem 4 (Answer and Construction)

Given A and B in the figure,

(a)    Tell what transformation A₁₂₀ B₉₀ is.  Answer: ______________

(b)   Construct (with straightedge and compass) the defining data for this transformation.

Problem 5 (Definition and short proof)

(a)    State the definition of an isometry. (Note: A definition is a complete sentence with just the definition -- no extra stuff that you know.)

(b)   It is said that isometries of the plane preserve angles.  What does this statement mean, precisely?

(c)    Prove your statement in (b), using the definition in (a).

Problem 6 (Construction)

The two segments AB and CD are congruent. 

Construct the defining data (invariant line m and glide vector EF) for a glide reflection G so that G(A) = C and G(B) = D.

Problem 7 (Short Answer)

The figure below is part of an infinite pattern of squares.  Let S = A₉₀ and T = C₉₀.  These questions are about the collection of all isometries that can be formed as products (compositions) from S and T (such as SSTTTSTTSTTT, etc.)

(a)    Which of the labeled points are centers of 90-degree rotations that are products formed from S and T? __________________________

(b)   Which of the labeled points (not already included in (a)) are centers of 180-degree rotations that are products formed from S and T? __________________________

(c)    Are there any unlabeled points inside square GJMP that are centers of 180-degree rotations that are products formed from S and T?  Yes or No? _____________

(d)   Which labeled points are images of point A by translations that are products formed from S and T? __________________________

(e)    Which labeled points are images of point A by glide reflections that are products formed from S and T? __________________________