Mathematics 138
Problem Set 4 due Wednesday, March 9, 2005
Note: tan 37o = 3/4 and tan 74o = 24/7 by definition.  Thus in problem 4, 0.96 = sin 74o.
Problems marked with * may be easier after the class on Monday, March 7.
1a. Determine what 3x3 matrix T represents the glide reflection that results from reflecting in the line 3x - 4y = 8, then translating 10 units up and to the right parallel to this line.
b. The matrix T suggests realizing the glide reflection by doing a reflection in a line through the origin, followed by a translation.  Draw a diagram showing how this can be accomplished as a product of three reflections, then apply the algorithm presented in the lecture notes to convert this product of three reflections back into a glide reflection (which should be the one that you started with!)
2. Express the Euclidean isometry described by the matrix S = as a rotation;
that is, specify the fixed point of the rotation and the angle.
b. The matrix S suggests an isometry that consists of a rotation about the origin, followed by a translation.  Draw a diagram showing how this can be accomplished as a product of four reflections, then apply the algorithm presented in the lecture notes to convert this product of four reflections back into a rotation (which should be the one that you started with!) 
3. Let ABC be an equilateral triangle.  Consider the following isometry:
a. Reflect in side AB.
b. Then reflect in side AC
c. Finally, reflect in side BC.
Being a product of an odd number of reflections, this must be either a reflection or a glide reflection.
Describe it in geometric terms.
4 a.The 3x3 matrix T represents a rotation.
a. Through what angle? Clockwise or counterclockwise?
b. What is its fixed point? (the coordinates are small integers)
c. Find the two square roots of T.  (This is easy if you think geometrically -- suppose the origin were at the fixed point?)
5. Consider two glide reflections
S has the x-axis as its axis, and there is a translation through 2a along the positive x-axis.
T has the y-axis as its axis, and there is a translation through 2b along the positive y-axis.
a. Write matrices for S and T, and compute the products ST and TS algebraically.
b. Compute the products ST and TS geometrically.  Remember that for a glide reflection,
you can do the translation and the reflection in either order.

6. A, B, and C are any three fixed non-collinear points in the plane. Consider the following three counterclockwise rotations.
Wa about A through an angle that would carry b into c.
Wb about B through an angle that would carry c into a. 
Wc about C through an angle that would carry a into b.
Suppose these are carried out in the order
listed. Note -- point B is fixed, so the second
rotation is about B, not about the point into
which B would be carried by the first rotation.
Describe the product WcWbWa geometrically.
It is a rotation: specify the angle and
the fixed point.
Hint: you may use the theorem of Euclid
that the angle bisectors of any triangle
are concurrent.

7a. Ryan, problem 27 on page 37.
b. Ryan, problem 34 on page 38. The "pencil" might be a parallel pencil, or it might be a concurrent pencil of lines that meet at a point P.
*8. Ryan, Problem 2 on page 67.
*9. Ryan, problem 7 on page 67.
*10. By applying the algorithm that Ryan uses to prove Theorem 8 on page 44, find the 2x2 matrix A and the vector b for the affine transformation T that carries
into  , into  , andinto 
*11. a. Ryan, problem 11 on page 67.
b. Find the 3x3 matrix that represents a central dilatation with magnification 2 and fixed point , followed by a 90 degree counterclockwise rotation about the origin.
c. Describe geometrically the transformation that results from a central dilatation with magnification 2 and fixed point , followed by a central dilatation with magnification 1/2 and fixed point .