United States of AmericaPA
High School Core Standards - Geometry Assessment Anchors

# 4.04 Dilations

Lesson

We've learned that similar triangles 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$3cm, $4$4cm and $5$5cm. 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$6cm, $8$8cm and $10$10cm, as shown below.

### 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.
2. A dilation factor: the ratio by which we increase/ decrease the shape. We calculate a dilation factor just like we would calculate the ratio of the sides in similar triangles.
Remember!

A dilation factor can increase or decrease the size of the new shape e.g. A dilation factor of $3$3 means the new shape will be $3$3 times as big, whereas a dilation 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 dilation factor, $k$k, has $k>1$k>1, the image will be larger than the preimage