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? After completing this Concept, you'll be able to use the corresponding sides of dilated figures to solve problems like this one.

### Watch This

### Guidance

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) |

#### Example A

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 B

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 C

\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*}.

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### Guided Practice

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.

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?

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.

**Answers:**

1. 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.

2. 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*}.

3. 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*}

### Explore More

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.