Translation of Geometric Objects

A translation is a transformation that moves every point of a figure the same distance in the same direction. More precisely, a translation along a vector v maps every point A in the plane onto its image A' such that the following statements hold true.

• The length of AA' and the magnitude of v are equal.
• Segment AA' and v are parallel.
• Vectors AA' and v have the same direction.

These three properties imply that the quadrilateral formed by A, A', the tip of v, and the tail of v is a parallelogram.

Since translations preserve side lengths and angle measures, they are rigid motions. Additionally, translations map lines onto parallel lines.