Exercise 2

This exercise investigates the composition of reflections in the plane.

1. Find the result of first reflecting across the line $y=0$ and then $y=x$. What familiar operation is the cumulative effect of this composition?

2. What happens if you compose the operations in the opposite order; that is, what happens if you first reflect across $y=x$ and then $y=0$? What familiar operation results?

3. What familiar geometric operation results if you first reflect across the line $y=x$ and then $y=-x$?

4. What familiar geometric operation results if you first rotate by $90^\circ$ and then reflect across the line $y=x$?

It is a general fact that the composition of two reflections results in a rotation through twice the angle from the first line of reflection to the second. We will investigate this more generally in this exercise