# Lesson 3

Measuring Dilations

• Let’s dilate polygons.

### 3.1: Dilating Out

Dilate triangle $$FGH$$ using center $$C$$ and a scale factor of 3.

### 3.2: All the Scale Factors

Here is a center of dilation and a triangle.

1. Measure the sides of triangle $$EFG$$ (to the nearest mm).
2. Your teacher will assign you a scale factor. Predict the relative lengths of the original figure and the image after you dilate by your scale factor.
3. Dilate triangle $$EFG$$ using center $$C$$ and your scale factor.
4. How does your prediction compare to the image you drew?
5. Use tracing paper to copy point $$C$$, triangle $$EFG$$, and your dilation. Label your tracing paper with your scale factor.
6. Align your tracing paper with your partner’s. What do you notice?

1. Dilate triangle $$FEG$$ using center $$C$$ and scale factors:
1. $$\frac12$$
2. 2
2. What scale factors would cause some part of triangle $$E'F'G'$$’ to intersect some part of triangle $$EFG$$?

### 3.3: What Stays the Same?

1. Dilate quadrilateral $$ABCD$$ using center $$P$$ and your scale factor.
2. Complete the table.
Ratio $$\frac{PA’}{PA}$$ $$\frac{PB’}{PB}$$ $$\frac{PC’}{PC}$$ $$\frac{PD’}{PD}$$
Value
3. What do you notice? Can you prove your conjecture?
4. Complete the table.
Ratio $$\frac{B’A’}{BA}$$ $$\frac{C’B’}{CB}$$ $$\frac{D’C’}{DC}$$ $$\frac{A’D’}{AD}$$
Value
5. What do you notice? Does the same reasoning you just used also prove this conjecture?

### Summary

We know a dilation with center $$P$$ and positive scale factor $$k$$ takes a point $$A$$ along the ray $$PA$$ to another point whose distance is $$k$$ times farther away from $$P$$ than $$A$$ is.

The triangle $$A’B’C’$$ is a dilation of the triangle $$ABC$$ with center $$P$$ and with scale factor 2. So $$A’$$ is 2 times farther away from $$P$$ than $$A$$ is, $$B’$$ is 2 times farther away from $$P$$ than $$B$$ is, and $$C’$$ is 2 times farther away from $$P$$ than $$C$$ is.

Because of the way dilations are defined, all of these quotients give the scale factor: $$\frac{PA’}{PA} = \frac{PB’}{PB} = \frac{PC’}{PC} = 2$$.

If triangle $$ABC$$ is dilated from point $$P$$ with scale factor $$\frac{1}{3}$$, then each vertex in $$A’’B’’C’’$$ is on the ray from P through the corresponding vertex of $$ABC$$, and the distance from $$P$$ to each vertex in $$A’’B’’C’’$$ is one-third as far as the distance from $$P$$ to the corresponding vertex in $$ABC$$.

$$\frac{PA’’}{PA} = \frac{PB’’}{PB} = \frac{PC’’}{PC} = \frac{1}{3}$$

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. In other words, If segment $$AB$$ is dilated from point $$P$$ with scale factor $$k$$, then the length of segment $$AB$$ is multiplied by $$k$$ to get the corresponding length of $$A’B’$$.

$$\frac{A’’B’’}{AB} = \frac{B’’C’’}{BC} = \frac{A’’C’’}{AC} = k$$.

Corresponding side lengths of the original figure and dilated image are all in the same proportion, and related by the same scale factor $$k$$.

### Glossary Entries

• dilation

A dilation with center $$P$$ and positive scale factor $$k$$ takes a point $$A$$ along the ray $$PA$$ to another point whose distance is $$k$$ times farther away from $$P$$ than $$A$$ is.

Triangle $$A'B'C'$$ is the result of applying a dilation with center $$P$$ and scale factor 3 to triangle $$ABC$$.

• scale factor

The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.