# 5.11 Dilations

Lesson

We've learned that similar polygons have all corresponding sides in the same ratio. So if a shape is enlarged or reduced, all the side lengths will increase or decrease in the same ratio. This enlargement or reduction is called a dilation. For example, let's say $\triangle ABC$ABC has side lengths of $3$3 cm, $4$4 cm and $5$5 cm. If it is dilated by a scale factor of $2$2 to produce $\triangle XYZ$XYZ, then $\triangle XYZ$XYZ will have side lengths of $6$6 cm, $8$8 cm and $10$10 cm, as shown below.

 $\frac{6}{3}$63​ $=$= $\frac{8}{4}$84​ $=$= $\frac{10}{5}$105​ $=$= $2$2

If any of these ratios were not equal to $2$2, then this would not be a dilation.

### Dilating a shape

Well, we need two things:

1. A center of dilation: a point from where we start the enlargement. This may be inside or outside the original shape, and for now we will only use the origin.
2. A scale factor: the ratio by which we increase/ decrease the shape. We calculate a scale factor just like we would calculate the ratio of the sides in similar triangles.
Remember!

A scale factor can increase or decrease the size of the new shape e.g. A scale factor of $3$3 means the new shape will have side lengths $3$3 times as big, whereas a scale factor of $\frac{1}{2}$12 means the new shape will be $\frac{1}{2}$12 as big as the original.

In general,

• If the scale factor, $k$k, has $k>1$k>1, the image will be larger than the preimage