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# Dilation in the Coordinate Plane

## Multiplication of coordinates by a scale factor.

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Practice Dilation in the Coordinate Plane
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Graphs of Dilations

Quadrilateral WXYZ\begin{align*}WXYZ\end{align*} has coordinates W (-5,-5), X (-2,0), Y (2,3),\begin{align*}W \ (\text{-}5, \text{-}5), \ X \ (\text{-}2, 0), \ Y \ (2, 3),\end{align*} and Z (-1,3).\begin{align*}Z \ (\text{-}1, 3).\end{align*} Draw the quadrilateral on the Cartesian plane.

Suppose the quadrilateral undergoes a dilation centered at the origin of scale factor 13.\begin{align*}\frac{1}{3}.\end{align*} What is  the resulting image?

### Graphs of Dilations

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 pre-image) to create a new figure (called the image). The scale factor, r, determines how much bigger or smaller the dilated image will be compared to the preimage.

In order to graph a dilation, use the center of dilation and the scale factor. Find the distance between a point on the preimage and the center of dilation. Multiply this length by the scale factor. The corresponding point on the image will be this distance away from the center of dilation in the same direction as the original point.

If you compare the length of a side on the preimage to the length of the corresponding side on the image, the length of the side on the image will be the length of the side on the preimage multiplied by the scale factor.

#### Draw the preimage and image for the following dilations and determine the scale factor:

Line AB¯¯¯¯¯¯¯¯\begin{align*}\overline{A B}\end{align*} drawn from (-4, 2) to (3, 2) has undergone a dilation about the origin to produce A(6,3)\begin{align*}A^\prime(-6, 3)\end{align*} and B(4.5,3)\begin{align*}B^\prime(4.5, 3)\end{align*}

scale factor=dilation image lengthpreimage lengthscale factor=10.57.0scale factor=32\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{10.5}{7.0} \\ & \mathrm{scale \ factor} = \frac{3}{2}\end{align*}

#### Determine the coordinates and the scale factor for the following dilations:

1. The diamond ABCD\begin{align*}ABCD\end{align*} undergoes a dilation about the origin to form the image ABCD\begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}

scale factor=dilation image lengthpreimage lengthscalefactor=7.213.61scalefactor=2\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale factor} = \frac{7.21}{3.61} \\ & \mathrm{scale factor} = 2\end{align*}

1. The diagram below undergoes a dilation about the origin to form the dilation image.

scalefactor=dilation image lengthpreimage lengthscale factor=2.0010.00scale factor=15\begin{align*}& \mathrm{scale factor }= \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale \ factor }= \frac{2.00}{10.00} \\ & \mathrm{scale \ factor }= \frac{1}{5}\end{align*}

### Examples

#### Example 1

Earlier, you were told that quadrilateral WXYZ\begin{align*}WXYZ\end{align*} has coordinates W (-5,-5), X (-2,0), Y (2,3),\begin{align*}W \ (\text{-}5, \text{-}5), \ X \ (\text{-}2, 0), \ Y\ (2, 3),\end{align*} and Z (-1,3).\begin{align*}Z \ (\text{-}1, 3).\end{align*} If the quadrilateral undergoes a dilation centered at the origin of scale factor 13,\begin{align*}\frac{1}{3},\end{align*}  what is the resulting image?

Test to see if the dilation is correct by determining the scale factor.

scale factor=dilation image lengthpreimage lengthscale factor=10.633.54scale factor=3\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{10.63}{3.54} \\ & \mathrm{scale \ factor} = 3\end{align*}

#### Example 2

Line ST¯¯¯¯¯¯¯\begin{align*}\overline{S T}\end{align*} drawn from (-3, 4) to (-3, 8) has undergone a dilation of scale factor 3 about the point A (1,6).\begin{align*}A \ (1, 6).\end{align*} Draw the preimage and image and properly label each.

scale factor=dilation image lengthpreimage lengthscale factor=12.004.00scale factor=3\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{12.00}{4.00} \\ & \mathrm{scale \ factor} = 3\end{align*}

#### Example 3

The polygon below has undergone a dilation about the origin with a scale factor of 53.\begin{align*}\frac{5}{3}.\end{align*} Draw the dilation image and properly label each point.

scale factor=dilation image lengthpreimage lengthscale factor=5.003.00scale factor=53\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{5.00}{3.00} \\ & \mathrm{scale \ factor} = \frac{5}{3}\end{align*}

#### Example 4

The triangle with vertices J (-5,2), K (-1,4),\begin{align*}J \ (\text{-}5, -2), \ K \ (\text{-}1, 4),\end{align*} and L (1,-3)\begin{align*}L \ (1, \text{-}3)\end{align*} has undergone a dilation of scale factor 12\begin{align*}\frac{1}{2}\end{align*} about the center point L.\begin{align*}L.\end{align*} Draw and label the dilation image and the preimage, then verify the scale factor.

scale factor=dilation image lengthpreimage lengthscale factor=7.213.61scale factor=12\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale \ factor} = \frac{7.21}{3.61} \\ & \mathrm{scale \ factor} = \frac{1}{2}\end{align*}

### Review

1. Dilate the above figure by a factor of 12\begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of 2\begin{align*}2\end{align*} about point D.\begin{align*}D.\end{align*}

1. Dilate the above figure by a factor of 3\begin{align*}3\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}\frac{1}{4}\end{align*} about point \begin{align*}C.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}2\end{align*} about point \begin{align*}A.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}2\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}D.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}3\end{align*} about point \begin{align*}D.\end{align*}

1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*}

To see the Review answers, open this PDF file and look for section 10.11.

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Color Highlighted Text Notes

### Vocabulary Language: English

Dilation

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

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be defined as $d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Mapping

Mapping is a procedure involving the plotting of points on a coordinate grid to see the behavior of a function.

Scale Factor

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