GeoProof — Dilation (Scaling) explorer

image point = O + k · (point − O) · every distance from O scales by k

Dilation (Scaling)

A dilation resizes a figure from a fixed centre O by a scale factor k. Each point P maps to P′ along the ray from O, at k times the distance:

P′ = O + k·(P − O) so OP′ = k·OP

O, P and P′ are always lined up (collinear) — the dashed rays show it.

What k does

k > 1 enlarges; 0 < k < 1 shrinks; k = 1 leaves the figure unchanged. (A negative k would also flip the figure through O.)

Similarity, not congruence

Unlike the rigid motions, a dilation changes size — so the image is similar to the original, not congruent. Angles are preserved, every length is multiplied by k, and area is multiplied by k². The image stays the same shape, just bigger or smaller.

The centre is fixed

Only O stays put. Put O at a vertex and that vertex's image stays glued to it while the rest scales away from it.

Where it's used

Dilations are the heart of similar figures, scale drawings and maps, and (combined with a rigid motion) every similarity transformation.