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)