a dilation
Every other motion so far has kept sizes exactly. A dilation deliberately breaks that rule: it is a uniform zoom, enlarging or shrinking a figure away from (or toward) a fixed centre, like a photocopier set to 150 percent or a projector throwing a small slide onto a big screen. The shape stays the same — all the angles and proportions are untouched — but every length is multiplied by a fixed number.
A dilation has two ingredients: a centre O and a scale factor k. It sends each point P to the point P' on the ray from O through P with |OP'| = |k| times |OP|. In coordinates, dilating about the origin by factor k sends (x, y) to (kx, ky). If k > 1 the figure grows; if 0 < k < 1 it shrinks; if k = 1 nothing moves; and a negative k also flips the figure through the centre to the opposite side (a scaling combined with a point reflection). The centre O is the one fixed point. Crucially, a dilation multiplies every length by |k|, so it multiplies area by k^2 — doubling a shape's dimensions quadruples its area, the square-cube intuition in two dimensions.
Because it preserves angles and shape but not length, a dilation is not an isometry — it is the simplest non-rigid transformation, and it is exactly what makes two figures similar rather than congruent. Composing a dilation with a rigid motion gives a similarity transformation, the broadest class that preserves shape. Dilations underlie scale models, maps, and the meaning of scale factor: a 1:50 architectural model is a dilation of the real building by k = 1/50, faithfully shaped but fifty times smaller in every linear measurement.
Dilate triangle A(1, 1), B(3, 1), C(1, 4) about the origin with scale factor k = 2. Multiply each coordinate by 2: A' = (2, 2), B' = (6, 2), C' = (2, 8). Each side doubled: |AB| = 2 became |A'B'| = 4. The new triangle is similar to the old, twice as large in every length and four times the area.
A uniform zoom from a centre: shape and angles kept, but every length is multiplied by k.
A dilation is not an isometry — it preserves shape and angle but changes length, which is exactly why it produces similar (not congruent) figures. Multiplying lengths by k multiplies area by k^2, so do not scale area by the same factor as length.