 # 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

### Outcomes

#### GEO-G.CO.2

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

#### GEO-G.SRT.1

Verify experimentally the properties of dilations given by a center and a scale factor.

#### GEO-G.SRT.1a

Verify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

#### GEO-G.SRT.1b

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.