The Art and Craft of Problem Solving

304 CHAPTER 8 GEOMETRY FOR AMERICANS

how to construct an equilateral triangle, right? (If not, do Problems 8.2.28 and

8.3.3 1 !)

• Construct a line perpendicular to this rotated segment that pases through its
  endpoint. This will be f4 , and its intersection with f 2 is the point B. •

That was fun and rather easy, but we won't always be so lucky. Sometimes we

will need to combine more than one rotation. What happens when we compose two

rotations? First, a simple lemma.

Fact 8.5.6 Let f be a line and let f' be its image under rotation by e about the center

C. Then the angle between f' and f is also e.

This is not at all surprising. We are asserting that LX PY = e in the picture below.

This follows from easy angle chasing. Do it!

,

, ,

, e ,

4-..' , ,

, ,

·c

This lemma should allow you to easily prove the following composition facts.

Fact 8.5.7 The composition of a rotation by a with a rotation by f3 (possibly about a

different center) is

• a rotation by a + f3, if a + f3 =I- 360 °.


• a translation (not the identity), if a + f3 = 360 ° and the centers are different.


• the identity transfonnation, if a + f3 = 360 ° and the centers are the same.
  Fact 8.5.8 When the composition of two rotations is another rotation, here is a way to
  find the center. Let RA,a denote rotation by a about A. Likewise, define the rotation
  RB,{3. Draw rays AX, BY so that BAX = a /2 and Y BA = f3 /2. The intersection of AX

and BY is the center of the rotation RB,{3 0 RA,a. (Remember that composition To S

means "do S, then T .")

Fact 8.5.9 The composition of a rotation and a translation (taken in either order) is

a rotation by the same angle, but with a different center. (Finding the center is an

exercise.)

We conclude our discussion of rotations with a special case of a wonderful classic

problem.

Example 8.5.10 Let XYZ be a triangle. Erect exterior isosceles triangles on the sides

XY, YZ, XZ, with vertex points A, B, C, respectively, such that LA = 60 °,LB= 120 °,LC=