Similarity Transformations

A similarity transformation maps a figure to a similar figure. It combines:

• A dilation (scaling by a factor k)
• Optionally: rigid motions (reflections, rotations, translations)

Dilation

A dilation centered at Point O with scale factor k maps every Point P to a Point P' such that OP' = k × OP.

• If k > 1: the image is an enlargement.
• If 0 < k < 1: the image is a reduction.
• If k = 1: the image is congruent to the original.

Coordinate Rule for Dilation

Dilation centered at the origin with scale factor k: (x, y) → (kx, ky)

Key Properties

• Dilations preserve Angle measures.
• Dilations preserve parallelism.
• Dilations multiply all lengths by k.
• Two figures are similar if and only if one can be mapped to the other by a similarity transformation.