### Dilation

A **transformation** is an operation that moves, flips, or changes a figure to create a new figure. Transformations that preserve size are ** rigid** and ones that do not are

**A**

*non-rigid.***dilation**makes a figure larger or smaller, but has the same shape as the original. In other words, the dilation is similar to the original. All dilations have a

**center**and a

**scale factor.**The center is the point of reference for the dilation (like the vanishing point in a perspective drawing) and scale factor tells us how much the figure stretches or shrinks. A scale factor is typically labeled \begin{align*}k\end{align*} and is always greater than zero. Also, if the original figure is labeled \begin{align*}\triangle ABC\end{align*}, for example, the dilation would be \begin{align*}\triangle A'B'C'\end{align*}. The ‘ indicates that it is a copy. This tic mark is said “prime,” so \begin{align*}A'\end{align*} is read “A prime.” A second dilation would be \begin{align*}A''\end{align*}, read “A double-prime.”

*If the dilated image is smaller than the original, then the scale factor is \begin{align*}0 < k < 1\end{align*}.*

*If the dilated image is larger than the original, then the scale factor is \begin{align*}k > 1\end{align*}.*

#### Dilating a Point

1. The center of dilation is \begin{align*}P\end{align*} and the scale factor is 3. Find \begin{align*}Q'\end{align*}.

If the scale factor is 3 and \begin{align*}Q\end{align*} is 6 units away from \begin{align*}P\end{align*}, then \begin{align*}Q'\end{align*} is going to be \begin{align*}6 \times 3 = 18\end{align*} units away from \begin{align*}P\end{align*}. Because we are only dilating a point, the dilation will be collinear with the original and center.

2. Using the picture above, change the scale factor to \begin{align*}\frac{1}{3}\end{align*}. Find \begin{align*}Q''\end{align*}.

Now the scale factor is \begin{align*}\frac{1}{3}\end{align*}, so \begin{align*}Q''\end{align*} is going to be \begin{align*}\frac{1}{3}\end{align*} the distance away from \begin{align*}P\end{align*} as \begin{align*}Q\end{align*} is. In other words, \begin{align*}Q''\end{align*} is going to be \begin{align*}6 \times \frac{1}{3} = 2\end{align*} units away from \begin{align*}P\end{align*}. \begin{align*}Q''\end{align*} will also be collinear with \begin{align*}Q\end{align*} and center.

#### Drawing a Dilation

\begin{align*}KLMN\end{align*} is a rectangle with length 12 and width 8. If the center of dilation is \begin{align*}K\end{align*} with a scale factor of 2, draw \begin{align*}K'L'M'N'\end{align*}.

If \begin{align*}K\end{align*} is the center of dilation, then \begin{align*}K\end{align*} and \begin{align*}K'\end{align*} will be the same point. From there, \begin{align*}L'\end{align*} will be 8 units above \begin{align*}L\end{align*} and \begin{align*}N'\end{align*} will be 12 units to the right of \begin{align*}N\end{align*}.

### Examples

#### Example 1

Find the perimeters of \begin{align*}KLMN\end{align*} and \begin{align*}K'L'M'N'\end{align*}. Compare this ratio to the scale factor.

The perimeter of \begin{align*}KLMN = 12 + 8 + 12 + 8 = 40\end{align*}. The perimeter of \begin{align*}K'L'M'N' = 24 + 16 + 24 + 16 = 80\end{align*}. The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.

#### Example 2

\begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}. If \begin{align*}P\end{align*} is the center of dilation, what is the scale factor?

Because \begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}, then \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}. The scale factor is the ratio of the sides. Since \begin{align*}\triangle ABC\end{align*} is smaller than the original, \begin{align*}\triangle DEF\end{align*}, the scale factor is going to be less than one, \begin{align*}\frac{12}{20} = \frac{3}{5}\end{align*}.

If \begin{align*}\triangle DEF\end{align*} was the dilated image, the scale factor would have been \begin{align*}\frac{5}{3}\end{align*}.

#### Example 3

Find the scale factor, given the corresponding sides. In the diagram, the **black** figure is the original and \begin{align*}P\end{align*} is the center of dilation.

Since the dilation is smaller than the original, the scale factor is going to be less than one. \begin{align*}\frac{8}{20}=\frac{2}{5}\end{align*}

### Review

In the two questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the **black** figure is the original and \begin{align*}P\end{align*} is the center of dilation.

- \begin{align*}k = 4\end{align*}
- \begin{align*}k = \frac{1}{3}\end{align*}

In the question below, find the scale factor, given the corresponding sides. In the diagram, the **black** figure is the original and \begin{align*}P\end{align*} is the center of dilation.

- Find the perimeter of both triangles in #1. What is the ratio of the perimeters?
What happens if \begin{align*}k = 1\end{align*}?*Writing*

** Construction** We can use a compass and straight edge to construct a dilation as well. Copy the diagram below.

- Set your compass to be \begin{align*}CG\end{align*} and use this setting to mark off a point 3 times as far from \begin{align*}C\end{align*} as \begin{align*}G\end{align*} is. Label this point \begin{align*}G'\end{align*}. Repeat this process for \begin{align*}CO\end{align*} and \begin{align*}CD\end{align*} to find \begin{align*}O'\end{align*} and \begin{align*}D'\end{align*}.
- Connect \begin{align*}G', O'\end{align*} and \begin{align*}D'\end{align*} to make \begin{align*}\triangle D'O'G'\end{align*}. Find the ratios, \begin{align*}\frac{D'O'}{DO}, \frac{O'G'}{OG}\end{align*} and \begin{align*}\frac{G'D'}{GD}\end{align*}.
- What is the scale factor of this dilation?
- Describe how you would dilate the figure by a scale factor of 4.
- Describe how you would dilate the figure by a scale factor of \begin{align*}\frac{1}{2}\end{align*}.

- The scale factor between two shapes is 1.5. What is the ratio of their perimeters?
- The scale factor between two shapes is 1.5. What is the ratio of their areas?
*Hint: Draw an example and calculate what happens.* - Suppose you dilate a triangle with side lengths 3, 7, and 9 by a scale factor of 3. What are the side lengths of the image?
- Suppose you dilate a rectangle with a width of 10 and a length of 12 by a scale factor of \begin{align*}\frac{1}{2}\end{align*}. What are the dimensions of the image?
- Find the areas of the rectangles in #14. What is the ratio of their areas?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 7.11.