Dilation

A dilation is, simply put, an enlargement or shrinking of an object. This is also known as a similitude, which means that the figure is getting bigger or smaller by some factor. A dilation is performed as follows: Suppose we have a plane figure. Without loss of generality, say it is a quadrilateral, $ABCD$ .

Now, choose a center of dilation, and name it $O$ . Choose an arbitrary factor of dilation $k$

Draw $OA^{\rightarrow}$ . Multiply the length by $k$ to get $OA'^{\rightarrow}$ . Do this for all points to get quadrilateral $A'B'C'D'$ .

In general, we say that the planar figure with vertices $a,b,c\dots,n$ dilated by a factor $k$ around a center $O$ results in the vertices $a',b',c'\dots,n'$ such that $k(OA^{\rightarrow})$ = $OA'^{\rightarrow}$ , $k(OB^{\rightarrow})$ = $OB'^{\rightarrow}$ , etc.

The resulting figures are similar (with congruence iff the factor is 1), and are oriented the same way, meaning any distances between the points are of the same ratio. For example, $\frac{OA}{OA'}=\frac{OB}{OB'}=\frac{OC}{OC'},\dots,=\frac{ON}{ON'}$ . The areas are in a ratio of $k^2$ .

Conversely, any two similar figures in a plane are dilations of each other. These figures are called homothetic figures. These figures are special, because if we drew lines through corresponding parts of the two figures, they would intersect at a common point $O$ , the center of dilation.

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Dilation