# What are some examples of dilation?

Feb 27, 2016

See below

#### Explanation:

Here is an example of dilation (also known as scaling).

Dilation is a transformation on the plane when there is a rule that transforms every point to its image as follows.

There is a center of dilation $O$ and factor $f \ne 0$.
Every point $A$ on the plane (except center$O$) is transformed into point $A '$ such that
(a) point $A '$ lies on line $O A$
(b) length $O A '$ equals to length of $O A$ multiplied by factor $f$
(if $f > 0$, points $A$ and $A '$ lie on the same side from center $O$; if $f < 0$, center $O$ should lie in between $A$ and $A '$)

Dilation preserves parallelism between lines.
Dilation results in all segments to change their length by the same factor $f$.
Dilation preserves the angles between lines.

Dilation transforms geometric figures into similar ones.
Actually, the definition of similarity is best approached from the position of dilation:
Two geometric objects, $\alpha$ and $\beta$, are called similar if there is a transformation of dilation (or scaling) defined by some center $O$ and some factor $f$ that transforms object $\alpha$ into object $\alpha '$, congruent to $\beta$.

For more properties of dilation (scaling) look into Web site Unizor dedicated to presentation of advanced math for high school. There you can find proofs of the above properties and numerous problems.