# How do you calculate the scale factor of a dilation?

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Dec 25, 2015

If a dilation) (or scaling) is given, it is assumed that its center and a factor are given, so we can construct an image of any point.

If center of dilation is point $O$ and factor is f≠0, any given point $A$ is transformed by a dilation into point $A '$ such that
(a) points $O$, $A$ and $A '$ are on the same line;
(b) if $f > 0$, points $A$ and $A '$ are on the same side from center $O$; if $f < 0$, point $O$ is in between $A$ and $A '$;
(c) Lengths of segments $O A '$ and $O A$ relate to each other at factor $| f |$, that is $| O A ' \frac{|}{|} O A | = | f |$

If these two parameters, the center and the factor, are not known, something must be given to determine them.

If a center $O$ is given, to determine a factor we need points $A$ and its image (the result of scaling) $A '$.
Then, knowing mutual position of points $O$, $A$ and $A '$ and lengths of segments $O A$ and $O A '$, we can determine a factor, its sign based on position of given points and its absolute value as the ratio of the lengths of segments $O A '$ and $O A$.

If even a center is not given, we need two pairs of points: sources $A$ and $B$ and their images $A '$ and $B '$.
Then we determine the center as an intersection of lines $A A '$ and $B B '$ and then follow the procedure above since a center is known now.

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