### Dilation

Two figures are similar if they are the same shape but not necessarily the same size. One way to create similar figures is by dilating. A dilation makes a figure larger or smaller but the new resulting figure has the same shape as the original.

**Dilation:** An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

Dilations have a **center** and a **scale factor**. The center is the point of reference for the dilation and the scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled \begin{align*}k\end{align*}. Only positive scale factors, k, will be considered in this text.

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

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

A dilation, or image, is always followed by a \begin{align*}'\end{align*}.

Label It |
Say It |
---|---|

\begin{align*}'\end{align*} | “prime” (copy of the original) |

\begin{align*}A'\end{align*} | “a prime” (copy of point \begin{align*}A\end{align*}) |

\begin{align*}A''\end{align*} | “a double prime” (second copy) |

What if you enlarged or reduced a triangle without changing its shape? How could you find the scale factor by which the triangle was stretched or shrunk?

### 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

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*}. The dilated image will be on the same line as the original image and center.

#### Example 4

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

Find \begin{align*}Q''\end{align*} using this new scale factor.

The scale factor is \begin{align*}\frac{1}{3}\end{align*}, so \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.

#### Example 5

\begin{align*}KLMN\end{align*} is a rectangle. If the center of dilation is \begin{align*}K\end{align*} and \begin{align*}k = 2\end{align*}, 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 \begin{align*}8\end{align*} 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*}.

### Review

For the given shapes, draw the dilation, given the scale factor and center.

- \begin{align*}k=3.5\end{align*}, center is \begin{align*}A\end{align*}

- \begin{align*}k=2\end{align*}, center is \begin{align*}D\end{align*}

- \begin{align*}k = \frac{3}{4}\end{align*}, center is \begin{align*}A\end{align*}

- \begin{align*}k = \frac{2}{5}\end{align*}, center is \begin{align*}A\end{align*}

In the four 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*}

- \begin{align*}k = 2.5\end{align*}

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

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

### Review (Answers)

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