Dilations and Scale Factor

A dilation is the transformation that makes a figure bigger or smaller without changing its shape. Pick a center point and a scale factor k. Every point P moves to a new point P' so that P' lies on the ray from the center through P, at a distance k times as far from the center as P was. Angles never change under a dilation because the figure is just stretched uniformly in every direction from one fixed point. That is why a projector image keeps every angle of the slide identical while every length grows by the same factor.

When the center is the origin, the mechanism becomes pure arithmetic: multiply both coordinates by k. A point (x, y) maps to (kx, ky). This matches the ray-and-distance idea exactly, since scaling both coordinates scales the distance from the origin by the same amount. A line y = mx + c dilated about the origin keeps its slope m untouched, because slope is a ratio of changes, and only the intercept gets multiplied by k. Lengths scale by k, but area scales by k squared, since area comes from two linear dimensions multiplied together. A scale factor of 1/2 shrinks each side to half, so the area shrinks to 1/4, not 1/2.

Dilation sits next to rigid motions like rotations and reflections in this unit, but it breaks the rule those transformations follow: it does not preserve distance. It does preserve angle measure and shape, which is why dilated figures are always similar to the original, never congruent unless k = 1. Negative scale factors push this idea further by sending points to the opposite side of the center entirely.