**Theorem 4.5**: A
translation is an
isometry.

**Proof**: Let* **T*((*x* , *y*) = (*x* - *x*_{0} , *y* -
*y*_{0}).* *Let* *(*x*_{1}, *y*_{1})* *and* *(*x*_{2}, *y*_{2})* *be two points in the
plane. Then by the
distance formula,

so the the translation satisfies the definition of an isometry.