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A dilation scales a figure by a scale factor k from a center of dilation. Each point moves along the ray from the center through the point, to k times its original distance from the center. If k > 1, the figure enlarges; if 0 < k < 1, it shrinks; if k < 0, it also reflects. Dilations preserve angle measures and shape (the image is similar to the preimage) but not distances (unless k = 1). For a dilation centered at the origin, (x, y) maps to (kx, ky).
Draw a figure and its dilation from a center point using rays and a scale factor. Verify that angles are preserved and sides are proportional. Practice with the coordinate rule for dilations centered at the origin. Explore non-origin centers. Connect dilations to similarity: two figures are similar if and only if one can be obtained from the other by a sequence of rigid motions and dilations.
You already know that similar triangles have equal angles and proportional sides — one is a scaled version of the other. A dilation is the transformation that makes this precise: it's the exact geometric operation that produces similarity. Given a center of dilation O and a scale factor k, every point P moves to a new point P' such that O, P, and P' are collinear, and OP' = k · OP. The point travels along the ray from O through P, stopping at k times its original distance from O. When k = 2, every point doubles its distance from O; the figure doubles in size but keeps its exact shape.
From the coordinate plane you know well, dilations centered at the origin are especially clean: the point (x, y) maps to (kx, ky). You're scaling both coordinates by the same factor. This is why dilations preserve angle measures — if you scale x and y by the same k, the ratio that determines any angle stays constant. What changes is distance: the distance between two points scales by |k|. This is the crucial contrast with the rigid motions (translations, rotations, reflections) you've seen: rigid motions preserve all distances and angles, so the image is congruent to the preimage. Dilations preserve angles but scale distances, so the image is similar to the preimage, not congruent (unless k = 1).
The sign and magnitude of k determine the character of the transformation. When k > 1, the figure enlarges. When 0 < k < 1, it shrinks — every point moves closer to the center, but the shape is intact. When k < 0, something more interesting happens: the point P' lands on the opposite side of O from P (since you're traveling a negative distance along the ray from O through P). This simultaneously scales and reflects through the center, producing a figure that is similar to the original but rotated 180°. The magnitude |k| still controls the size change.
For dilations with non-origin centers, you can always translate so that the center lands at the origin, apply the coordinate rule, then translate back — connecting your knowledge of translations directly. Two figures are similar if and only if one can be mapped to the other by a sequence of dilations and rigid motions. So dilations are the missing piece that completes the similarity story: congruence transformations (rigid motions alone) plus dilation gives the full family of similarity transformations. Every similar-triangles result from your AA similarity work corresponds to a concrete dilation you could explicitly construct.