How do you locate the center of a dilation?

1 Answer
Dec 25, 2015

Answer:

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!=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 #OA'# and #OA# relate to each other at factor #|f|#, that is #|OA'|/|OA| = |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 #BB'#. 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 #OA# and #OA'#.