Using the applet below, move the point P to see what happens when it is reflected over line n (to point P') and then over line m (to point P''). Alter line n by moving point Q. What is the combined effect of the two reflections?

Use the applet below to investigate the result of composing two reflections across lines that are parallel. What might you conjecture for this case?

Let A be an arbitrary point in the plane and consider its reflection over line m, followed by a reflection over line n. By definition of (mirror) reflection, the first reflection will transform A to a point, A', that lies on the perpendicular to m passing through A; and the distance, d[sub]1[/sub], to m (along that perpendicular line) will be the same for both points. [br][br]Because lines n and m are parallel, the reflection of point A' across n will transform A' along that same perpendicular line to point A'', where A' and A'' have the same distance, d[sub]2[/sub], to line n. [br][br]Note that d[sub]1[/sub]+d[sub]2[/sub] is the distance between lines m and n. So every point in the plane is translated in the same direction (perpendicular to lines m and n) by the same amount (twice the distance between the lines).

Composing two reflections yields a rotation or a translation. So, what happens when you compose two rotations? The app below shows the results of rotating the Hokie Bird through angle A and then angle B. It also shows what happens to the Hokie Bird when reflected over the two lines that make up each rotation. Try moving the original Hokie bird (the one with blue dots) and see the effects of each reflection and rotation, as well as their compositions.

The applet below represents a rotation of the plane, about point B, by angle ABC, followed by a translation of the plane, as indicated by the direction and length of DE. Show that their composition can be represented by the composition of two reflections.

The applet below represents a composition of reflections of the plane, over four lines: a, b, c, and d. Show that we can "collapse" that composition down to a composition of reflections over two lines