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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 #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.

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 #OA# and #OA'#, 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 #OA'# and #OA#.

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 #BB'# and then follow the procedure above since a center is known now.

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