How does a dilation affect a figure on a coordinate plane?

1 Answer
Mar 10, 2016

Answer:

Assuming the center of dilation is at point #{0,0}# on the coordinate plane and a factor of dilation #f#, a point #A{x,y}# will be transformed into point #A'{fx,fy}#.
See more details below.

Explanation:

Dilation or scaling is the transformation of the plane 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|#.

Using coordinates, the above properties can be expressed in the following form.
Assuming the center of dilation is at point #{0,0}# on the coordinate plane and a factor of dilation #f#, a point #A{x,y}# will be transformed into point #A'{fx,fy}#.
If the center of dilation is at point #C{p,q}#, the point #A{x,y}# will be transformed by dilation into #A'{p+f(x-p),q+f(y-q)}#

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