How does dilation affect the length of line segments?

Dilation of segment ##AB## is a segment ##A'B'##, where ##A'## is an image of point ##A## and ##B'## is an image of point ##B##. The length is transformed as ##|A'B'| = f*|AB|##, where ##f## is a factor of dilation

Dilation or scaling is the transformation of the two-dimensional plane or three-dimensional space according to the following rules:

(a) There is a fixed point ##O## on a plane or in space that is called the center of scaling.

(b) There is a real number ##f!=0## that is called the factor of scaling.

(c) The transformation of any point ##P## into its image ##P'## is done by shifting its position along the line ##OP## in such a way that the length of ##OP'## equals to the length of ##OP## multiplied by a factor ##|f|##, that is ##|OP'| = |f|*|OP|##. Since there are two candidates for point ##P'## on both sides from center of scaling ##O##, the position is chosen as follows: for ##f>0## both ##P## and ##P'## are supposed to be on the same side from center ##O##, otherwise, if ##f<0##, they are supposed to be on opposite sides of center ##O##.

It can be proven that the image of a straight line ##l## is a straight line ##l'##. Segment ##AB## is transformed into a segment ##A'B'##, where ##A'## is an image of point ##A## and ##B'## is an image of point ##B##.Dilation preserves parallelism among lines and angles between them. The length of any segment ##AB## changes according to the same rule above: ##|A'B'| = f*|AB|##.

The above properties and other important details about transformation of scaling can be found on Unizor