Theorem 4.5: A translation is an isometry.

Proof: Let   T((x , y) = (x - x0 , y - y0).   Let   (x1, y1)   and   (x2, y2)   be two points in the plane. Then by the distance formula,

|T((x1, y1))T((x2, y2))|

= |(x1 - x0, y1 - y0)(x2 - x0, y2 - y0)|

= | (x1, y1), (x2, y2) |

so the the translation satisfies the definition of an isometry.

next theorem (4.6)