Rotate a Figure Using Reflection

By Mark Ryan

A rotation is what you’d expect—it’s a geometric transformation in which the pre-image figure rotates or spins to the location of the image figure.

With all rotations, there’s a single fixed point—called the center of rotation—around which everything else rotates. This point can be inside the figure, in which case the figure stays where it is and just spins. Or the point can be outside the figure, in which case the figure moves along a circular arc (like an orbit) around the center of rotation. The amount of turning is called the rotation angle.

You can achieve a rotation with two reflections. The way this works is a bit tricky to explain (and the mumbo-jumbo in the following theorem might not help much), so check out the figure to get a better handle on this idea.

Two reflections make a rotation.

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

  • Pass through the center of rotation
  • Form an angle half the measure of the rotation angle

In the figure, you can see that pre-image triangle RST has been rotated counterclockwise 70 degrees to image triangle R’S’T’. This rotation can be produced by first reflecting triangle RST over line l1 and then reflecting it again over l2. The angle formed by l1 and l2, 35 degrees, is half of the angle of rotation.