# 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.

- Measure the sides of triangle \(EFG\) (to the nearest mm).
- 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.
- Dilate triangle \(EFG\) using center \(C\) and your scale factor.
- How does your prediction compare to the image you drew?
- Use tracing paper to copy point \(C\), triangle \(EFG\), and your dilation. Label your tracing paper with your scale factor.
- Align your tracing paper with your partner’s. What do you notice?

- Dilate triangle \(FEG\) using center \(C\) and scale factors:
- \(\frac12\)
- 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?

- Dilate quadrilateral \(ABCD\) using center \(P\) and your scale factor.
- Complete the table.
Ratio \(\frac{PA’}{PA}\) \(\frac{PB’}{PB}\) \(\frac{PC’}{PC}\) \(\frac{PD’}{PD}\) Value - What do you notice? Can you prove your conjecture?
- Complete the table.
Ratio \(\frac{B’A’}{BA}\) \(\frac{C’B’}{CB}\) \(\frac{D’C’}{DC}\) \(\frac{A’D’}{AD}\) Value - 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.