How do you locate the center of a dilation?

Dec 25, 2015

This question assumes a task of finding a center of dilation by something that is given.
See below the details about what should be given and how it can be used.

Explanation:

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 $f \ne 0$, 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.
The minimum required to determine them is a source and an image of two different points.

Assume we have two points $A$ and $B$ and their images $A '$ and $B '$ as a result of dilation.
Since center of dilation $O$ must lie on the same line as points $A$ and $A '$, we can construct this line $A A '$ and state that center $O$ is located on it.
Analogously, center $O$ must lie on line $B B '$. Let's construct it as well.
The intersection of these two lines (and they must intersect since we know that center $O$ belongs to both) is our center of dilation.

Incidentally, we can find a factor of dilation since we know the relative position of points $O$, $A$ and $A '$ and can measure the length of segments $O A$ and $O A '$.