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

Mar 10, 2016

Assuming the center of dilation is at point $\left\{0 , 0\right\}$ on the coordinate plane and a factor of dilation $f$, a point $A \left\{x , y\right\}$ will be transformed into point $A ' \left\{f x , f y\right\}$.
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 $f \ne 0$ 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 $O P$ in such a way that the length of $O P '$ equals to the length of $O P$ multiplied by a factor $| f |$, that is $| O P ' | = | f | \cdot | O P |$. 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 $A B$ 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 $A B$ changes according to the same rule above: $| A ' B ' | = f \cdot | A B |$.

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

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