(in description)

Question

Lines l and m intersect at point Q forming an 80 degree angle. Point P is reflected in l, followed by the reflection in m. Describe the location of the image P''

Answer 1

A rotation is equivalent to two reflections over two lines that:

1. Pass through the center of rotation.
2. Intersect at an angle that is 1/2 the angle of rotation.

That means point Q is the center of rotation. Envision a circle with center at Q with a radius QP. The location of P'' will be on that circle where the angle between rays QP and QP'' is 160°.

This Desmos graph depicts a specific example, where line l is the y-axis and line m forms an 80° angle with the y-axis. P is rotated 160° to P'' and the reflections across l and m are depicted by showing that line l is a perpendicular bisector of PP' and line m is a perpendicular bisector of P'P''. The center of rotation is the origin.

desmos.com/calculator/ceyzcgywln

It would be possible to generalize the above:

a. center of rotation is somewhere besides the origin.

b. Lines l and m from an angle of 80° and intersect at the center of rotation.