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# Dilation

## Larger or smaller version of a figure that preserves its shape.

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Dilations

Which one of the following figures represents a dilation? Explain.

### Watch This

First watch this video to learn about dilations.

CK-12 Foundation Chapter10DilationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10DilationsB

### Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A dilation is a type of transformation that enlarges or reduces a figure (called the preimage) to create a new figure (called the image). The scale factor, r, determines how much bigger or smaller the dilation image will be compared to the preimage. The figure below shows that the image A\begin{align*}A^\prime\end{align*} is a dilation by a scale factor of 2.

Dilations also need a center point. The center point is the center of the dilation. You use the center point to measure the distances to the preimage and the dilation image. It is these distances that determine the scale factor.

#### Example A

Describe the dilation in the diagram below. The center of dilation is point H\begin{align*}H\end{align*}.

Solution:

Compare the lengths of corresponding sides to determine the scale factor. IJ¯¯¯¯\begin{align*}\overline{IJ}\end{align*} is 2 units long and IJ¯¯¯¯¯¯\begin{align*}\overline{I^\prime J^\prime}\end{align*} is 6 units long. 62=3\begin{align*}\frac{6}{2}=3\end{align*}, so the scale factor is 3. Therefore, the center point H is used to dilate IJK\begin{align*}\triangle IJK\end{align*} to IJK\begin{align*}\triangle I^\prime J^\prime K^\prime\end{align*} by a factor of 3.

#### Example B

Using the measurement below and the scale factor, determine the measure of the dilated image.

mAB¯¯¯¯¯r=15 cm=13

Solution: You need to multiply the scale factor by the measure of AB\begin{align*}AB\end{align*} in order to find the measurement of the dilated image AB\begin{align*}A^\prime B^\prime\end{align*}.

mAB¯¯¯¯¯¯¯=(r)mAB¯¯¯¯¯\begin{align*}m \overline{A^\prime B^\prime} = (r) m \overline{A B}\end{align*}

mAB¯¯¯¯¯¯¯=13(15)mAB¯¯¯¯¯¯¯=5 cm

#### Example C

Using the measurement below and the scale factor, determine the measure of the preimage.

mHI¯¯¯¯¯¯¯r=24 cm=2

Solution: Here, you need to divide the scale factor by the measurement of HI\begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage HI\begin{align*}HI\end{align*}.

mHI¯¯¯¯¯¯¯=(r)mHI¯¯¯¯¯\begin{align*}m \overline{H^\prime I^\prime} = (r) m \overline{H I}\end{align*}

24=2mHI¯¯¯¯¯mHI¯¯¯¯¯=242mHI¯¯¯¯¯=12 cm

#### Concept Problem Revisited

Which one of the following figures represents a dilation? Explain.

You know that a dilation is a transformation that produces an image of the same shape but larger or smaller. Both of the figures above represent objects that involve dilations. In the figure with the triangles, the scale factor is 3.

The second figure with the squares also represents a dilation. In this figure, the center point \begin{align*}(3,-2)\end{align*} is used to dilate the small square by a factor of \begin{align*}2\end{align*}.

### Guided Practice

1. Using the measurement below and the scale factor, determine the measure of the preimage.

2. Describe the dilation in the diagram below.

3. Quadrilateral \begin{align*}STUV\end{align*} has vertices \begin{align*}S(-1, 3), T(2, 0), U(-2, -1),\end{align*} and \begin{align*}V(-3, 1)\end{align*}. The quadrilateral undergoes a dilation about the origin with a scale factor of \begin{align*}\frac{8}{5}\end{align*}. Sketch the preimage and the dilation image.

1. Here, you need to divide the scale factor by the measurement of \begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage \begin{align*}HI\end{align*}.

\begin{align*}m \overline{T^\prime U^\prime} = |r| m \overline{T U}\end{align*}

2. Look at the diagram below:

In the figure, the center point \begin{align*}D\end{align*} is used to dilate the \begin{align*}A\end{align*} by a factor of \begin{align*}\frac{1}{2}\end{align*}.

3. Look at the diagram below:

### Explore More

Find the measure of the dilation image given the following information:

Find the measure of the preimage given the following information:

Describe the following dilations:

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.10.

### Vocabulary Language: English

Dilation

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.

A quadrilateral is a closed figure with four sides and four vertices.
Ratio

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.
Scale Factor

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.
Transformation

Transformation

A transformation moves a figure in some way on the coordinate plane.
Vertex

Vertex

A vertex is a point of intersection of the lines or rays that form an angle.
Rigid Transformation

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.