All transformations fall into one of four categories: translation, rotation, reflection, and glide reflection. Samples of these transformations are shown in the picture below, which was taken from the Geometry Center's Technology in the Geometry Classroom course materials.

A translation moves every point in the plane according to some vector of translation. A rotation turns every point in the plane about a specified center by some specific angle. A glide reflection first translates the plane by some vector then reflects in a mirror parallel to the vector.

The computer programs "Tangram" and "Geometer's Sketchpad" are useful for providing hands-on experience of transformations. Use the Geometer's Sketchpad to answer the following questions:

1. What happens when you reflect an object across first one mirror, then another? Check your answer. Extra credit: what happens if you repeat this process?
2. If you perform two transformations one after another, you get a third, new transformation. Make up a "multiplication table" describing what transformations can arise from each possible combination of two transformations. Extra credit: pick one or two specific transformations. See what happens if you apply them repeatedly to an object.
3. If one point or image is the reflection of another, how can you find the line of mirror symmetry?
4. If one object is a rotated copy of another object, how can you find the center and angle of rotation? Extra credit: when is it impossible to answer this question?
5. Given two triangles, one of which is a transformed copy of the other, how can you find three reflections whose composition superimposes the first on the second? (HINT: match up corresponding pairs of vertices.) Extra credit: can you extend this to a way to send an arbitrary object onto a transformed image of itself using three or fewer reflections? If you can, then any transformation can be described as a composition of three or fewer reflections!