Why do all transformation preserve distance

Well, a distance-preserving transformation is called a rigid motion, and the name suggests that it moves the points of the plane around in a rigid fashion.

A transformation is distance-preserving if the distance between the images of any two points and the distance between the two original points are equal.

If that's confusing, I get it; basically if you transform something, the points from the transformation are image points. Take the distance from a pair of image points, and take the distance from a pair of original points, and they should be the same for a rigid motion.

I keep emphasizing this b/c not all transformations preserve distance; a dilation can grow or shrink things. But if you didn't go over dilations, don't say nothin XD

You're gonna need to post the table