# Which one of these is not an isometry?

A. a translation B. a 30 degrees rotation about the orgin followed by a flection in the x-axis C. a dilation D. a reflection in the line y = -x followed by a 160 degrees rotation about the point (1, -2)

An isometry is when a new spatial map is created from an old spatial map, while still preserving the distance between elements in the map.

Therefore

A translation is an isometry because all elements in the map are moved an equal amount.

A rotation is an isometry because all elements in the map are rotated an equal amount, and a flexion is an isometry because all elements in the map are bent around the point of flexion equally.

Consequently,

A, B, and D are all isometries as they involve translation, rotation, and flexion, all of which preserve the distance between elements in the map.

This leaves us with C. dilation.  By the process of elimination it must not be an isometry.  This is true because a dilation involves expanding the map outwards from the origin of the dilation.  In the process of doing so, the distance between the elements in the map increases, and therefore, a dilation cannot be an isometry