Assignment 9 (Due Fri, 12/5)

Note: This the last assignment to be turned in. There will be additional practice problems suggested for study as well.

Reading Assignment for Weeks 9-10

In this section reading Brown and looking at the problems and thinking about connections is even more important than usual.

9.1 Formula for reflections in parallel lines (20 points)

Given 4 parallel lines k₁, k₂, k₃, k₄ it is possible to choose an (x,y) (Cartesian) coordinate system so that the x-axis m is perpendicular to these lines and the y-axis n is parallel to the lines. In this case, the equations of the lines are x = a₁, x = a₂, x = a₃, x = a₄. Let the reflections in these lines be denoted by R₁, R₂, R₃, R₄.

Using these equations for the lines, answer these questions. Since this is a long problem, please organize and write your work and answers clearly and readably for full credit.

Part 1

• Start with the a formula for the reflection P' of the point P = (x,y) in line k₁.
  This formula is P' = (2a₁ - x,y). This is just a new use for the point reflection formula from before.

Part 2

• Write down a formula for the reflection P'' of the point P' in line k2 in terms of x and y. This gives a formula for P'' = R₂R₁(P).
• Based on the formula, tell what transformation is R₂R₁, i.e., tell what kind of isometry (using the terminology for names of kinds of isometries) and give a geometric description (in turns of mirror lines, or centers, or angles, or vectors, or whatever is appropriate)

Part 3

• Write down a formula for the reflection P''' of the point P'' in line k3 in terms of x and y. This gives a formula for P''' = R₃R₂R₁(P).
• Based on the formula, tell what transformation is R₃R₂R₁, i.e., tell what kind of isometry (using the terminology for names of kinds of isometries) and give a geometric description (in turns of mirror lines, or centers, or angles, or vectors, or whatever is appropriate)

Part 4

• Write down a formula for the reflection P'''' of the point P''' in line k4 in terms of x and y. This gives a formula for P'''' = R₄R₃R₂R₁(P).
• Based on the formula, tell what transformation is R₄R₃R₂R₁, i.e., tell what kind of isometry (using the terminology for names of kinds of isometries) and give a geometric description (in turns of mirror lines, or centers, or angles, or vectors, or whatever is appropriate)
• Since R₄R₃R₂R₁ = R₄(R₃R₂R₁) is the product of two isometries, R₄ and R₃R₂R₁, tell what kind of isometries these two are and how the nature of R₄R₃R₂R₁ follows from one of the parts above.
• Since R₄R₃R₂R₁ = (R₄R₃)(R₂R₁) is the product of two isometries, R₄R₃ and R₂R₁, tell what kind of isometries these two are. Write the formulas for each of these isometries. What is the general theorem about the composition of two isometries of this kind? How does that show up in the formulas?

Problem 9.2 (Composition of a line reflection and a rotation)

(I) Draw a line AB. Let M be reflection in line AB and let R be rotation by 45 degrees (counterclockwise as always) with center A.

(a) Tell in words precisely what kind of isometry (a single name) is the isometry T = MR.

(b) In this figure, construct the geometric defining data for T. (Construct the angles, etc. using straightedge and compass.

(II) With the same line AB, draw a point C somewhere not on the line. Let S be rotation by 45 degrees (counterclockwise as always) with center C.

(c) Tell in words precisely what kind of isometry (a single name) is the isometry U = MS.

(d) In this figure, construct the geometric defining data for U. (Construct the angles, etc. using straightedge and compass.

Problem 9.3 (Composition of two rotations)

Draw two points A and B. Then if R is rotation by 60 degrees with center A and S is rotation by 180 degrees with center B, then RS is also a rotation, as is SR.

a) Construct the point C that is the center of rotation of RS. Write down the angle of rotation. Tell what are the angles of the triangle ABC.

b) In the same figure, construct the point D that is the center of rotation of SR. Write down the angle of rotation. Tell what are the angles of the triangle ABD.

c) Construct the center E of the rotation S(RS). What is the angle of rotation of this isometry? What is the shape of the triangle CBE?

Problem 9.4 (Symmetry Short Answer)

The figure below is part of an infinite pattern of squares.  Let S = A90 and T = C180.  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)    Which labeled points are images of point A by translations that are products formed from S and T? _____________________

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

Problem 9.5

On graph paper, let A = (0,0), B = (0,5), C = (2, 2), D = (5, 6).

• Check that the segments AB and CD are congruent.
• Construct with straightedge and compass the center Q of a rotation R that takes A to C and B to D.
• Construct with straightedge and compass the invariant line FG and glide vector FG of a glide reflection that takes A to C and B to D.