Dilations & Similarity Transformations

A dilation resizes a figure from a fixed centre by a scale factor k. From the origin, the rule is simply:

(x, y) → (kx, ky)

If k > 1 the figure grows; if 0 < k < 1 it shrinks. A negative k also flips it through the centre.

Dilations are the odd one out. They preserve angles and parallelism, so the shape stays the same, but they change every length. The image is therefore similar to the original, not congruent — unless k = 1.

Because angles survive, slope survives too: a dilated line is parallel to the original, or is the same line if it passes through the centre.

Combining rigid motions with a dilation gives a similarity transformation, and that composition is exactly what "similar" means: two figures are similar when some sequence of rigid motions plus a dilation maps one onto the other.

A dilation with centre C and scale factor k ≠ 0 maps each point P to P′ on ray CP with CP′ = |k|·CP, and on the opposite ray when k < 0. Centred at the origin the rule is (x,y) → (kx, ky); centred at (a,b) it is (a + k(x−a), b + k(y−b)).

Dilations preserve angle measure, parallelism, collinearity and betweenness, but scale distances by |k|, areas by k² and volumes by k³. They are therefore not isometries, and the image is similar rather than congruent.

A line through the centre maps to itself; any other line maps to a parallel line. This is the geometric content of the Side-Splitter Theorem and the reason slope is preserved.

A similarity transformation is a composition of rigid motions with a dilation, and two figures are similar exactly when such a transformation maps one to the other. This transformational definition matches the classical angle-and-proportion definition but is easier to apply and generalises to figures that are not polygons.