What Is a Dilation?

How the factor controls the image

Unlike rigid motions, dilation does not usually preserve length. What it does preserve is angle measure and the shape relationship that makes the original and image similar figures.

This page keeps the center, the original figure, and the scaled image on the same board so you can see that dilation is controlled by rays and ratio, not by a loose resizing gesture.

With scale factor 2, a point 3 units from the center lands 6 units away on the same ray. With scale factor one half, it lands halfway toward the center. Negative scale factors add a turn through the center, so the sign deserves attention rather than being treated like an ordinary positive resize.

• A dilation enlarges or shrinks a figure by a scale factor.
• If the scale factor is greater than 1, the image is an enlargement; if it is between 0 and 1, the image is a reduction.
• Corresponding points lie on the same ray from the center of dilation in the usual central-dilation model.
• Dilation preserves angle measure and parallelism, but not absolute side length unless the scale factor is 1.

Where dilations help

• Use dilation when studying similar figures and scale drawings.
• Use it on the coordinate plane when image points are generated from a center and a scale factor.
• Use it in real-world resizing problems such as maps, blueprints, and digital zoom-style enlargement.

Dilation checks to make

• Do not treat dilation as a rigid motion; the figure usually changes size.
• Do not forget the center of dilation, because the same scale factor from a different center gives a different image.
• Do not confuse the scale factor with an amount added to side lengths; dilation multiplies distances from the center.