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Geometry: Dilations

Dilations

A dilation is a transformation that changes the length of all line segments by the same proportion. A dilation does not change the shape of a figure, but it can change the size. Because the size of the figure changes, dilations are not isometries. Because the shape is invariant under a dilation, the original figure and its image are similar. By “similar” I mean the mathematical notion of similarity: Angles are congruent and sides are proportional.

If the proportion involved in the dilation is equal to one, then no noticeable change will have occurred. That would be the identity dilation. If the proportion is less than one, the figure will shrink. If the proportion is greater than one, the figure will grow.

Solid Facts

A dilation is a transformation that changes the length of all line segments by the same proportion.

Just as you saw with isometries, collinearity, between-ness and angle measures are invariant under a dilation. You saw that any two congruent triangles are related to each other by a sequence of isometries. It turns out that any two similar triangles are also related to each other by a sequence of isometries and a dilation. In other words, if two triangles are similar, then you can make one from the other using only isometries to move it around and a dilation to change the size.

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Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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