### Dilation in the Coordinate Plane

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 such that the new image 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*}.

To dilate something in the coordinate plane, multiply each coordinate by the scale factor. This is called **mapping .** For any dilation the mapping will be \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}. In this text, the center of dilation will always be the origin.

What if you were given the coordinates of a figure and were asked to dilate that figure by a scale factor of 2? How could you find the coordinates of the dilated figure?

### Examples

For Examples 1 and 2, use the following instructions:

Given \begin{align*}A\end{align*} and the scale factor, determine the coordinates of the dilated point, \begin{align*}A'\end{align*}. You may assume the center of dilation is the origin. Remember that the mapping will be \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}.

#### Example 1

\begin{align*}A(-4, 6), k = 2\end{align*}

\begin{align*}A'(-8, 12)\end{align*}

#### Example 2

\begin{align*}A(9, -13), k = \frac{1}{2}\end{align*}

\begin{align*}A'(4.5, -6.5)\end{align*}

#### Example 3

Quadrilateral \begin{align*}EFGH\end{align*} has vertices \begin{align*}E(-4, -2), F(1, 4), G(6, 2)\end{align*} and \begin{align*}H(0, -4)\end{align*}. Draw the dilation with a scale factor of 1.5.

Remember that to dilate something in the coordinate plane, multiply each coordinate by the scale factor.

For this dilation, the mapping will be \begin{align*}(x, y) \rightarrow (1.5x, 1.5y)\end{align*}.

\begin{align*}& E(-4, -2) \rightarrow (1.5(-4), 1.5(-2)) \rightarrow E'(-6, -3)\\ & F(1, 4) \rightarrow (1.5(1), 1.5(4)) \rightarrow F'(1.5, 6)\\ & G(6, 2) \rightarrow (1.5(6), 1.5(2)) \rightarrow G'(9,3)\\ & H(0, -4) \rightarrow (1.5(0),1.5(-4)) \rightarrow H'(0, -6)\end{align*}

In the graph above, the blue quadrilateral is the original and the red image is the dilation.

#### Example 4

Determine the coordinates of \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle A' B' C'\end{align*} and find the scale factor.

The coordinates of the vertices of \begin{align*}\triangle ABC\end{align*} are \begin{align*}A(2, 1)\end{align*}, \begin{align*}B(5, 1)\end{align*} and \begin{align*}C(3, 6)\end{align*}. The coordinates of the vertices of \begin{align*}\triangle A'B'C'\end{align*} are \begin{align*}A'(6, 3), B'(15, 3)\end{align*} and \begin{align*}C'(9, 18)\end{align*}. Each of the corresponding coordinates are three times the original, so \begin{align*}k = 3\end{align*}.

#### Example 5

Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example B.

\begin{align*}& \underline{\triangle ABC} && \underline{\triangle A'B'C'}\\ & AB = \sqrt{(2-5)^2 + (1-1)^2} = \sqrt{9} =3 && A'B' = \sqrt{(6-15)^2 + (3-3)^2} = \sqrt{81} = 9\\ & AC = \sqrt{(2-3)^2 + (1-6)^2} = \sqrt{26} && A'C' = \sqrt{(6-9)^2+(3-18)^2} = 3 \sqrt{26}\\ & CB = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{29} && C'B' = \sqrt{(9-15)^2 +(18-3)^2} = 3 \sqrt{29}\end{align*}

From this, we also see that all the sides of \begin{align*}\triangle A'B'C'\end{align*} are three times larger than \begin{align*}\triangle ABC\end{align*}.

### Review

Given \begin{align*}A\end{align*} and \begin{align*}A'\end{align*}, find the scale factor. You may assume the center of dilation is the origin.

- \begin{align*}A(8, 2), A'(12, 3)\end{align*}
- \begin{align*}A(-5, -9), A'(-45, -81)\end{align*}
- \begin{align*}A(22, -7), A(11, -3.5)\end{align*}

The origin is the center of dilation. Draw the dilation of each figure, given the scale factor.

- \begin{align*}A(2, 4), B(-3, 7), C(-1, -2); k = 3\end{align*}
- \begin{align*}A(12, 8), B(-4, -16), C(0, 10); k = \frac{3}{4}\end{align*}

** Multi-Step Problem** Questions 6-9 build upon each other.

- Plot \begin{align*}A(1, 2), B(12, 4), C(10, 10)\end{align*}. Connect to form a triangle.
- Make the origin the center of dilation. Draw 4 rays from the origin to each point from #21. Then, plot \begin{align*}A'(2, 4), B'(24, 8), C'(20, 20)\end{align*}. What is the scale factor?
- Use \begin{align*}k =4\end{align*}, to find \begin{align*}A''B''C''\end{align*}. Plot these points.
- What is the scale factor from \begin{align*}A'B'C'\end{align*} to \begin{align*}A''B''C''\end{align*}?

If \begin{align*}O\end{align*} is the origin, find the following lengths (using 6-9 above). Round all answers to the nearest hundredth.

- \begin{align*}OA\end{align*}
- \begin{align*}AA'\end{align*}
- \begin{align*}AA''\end{align*}
- \begin{align*}OA'\end{align*}
- \begin{align*}OA''\end{align*}
- \begin{align*}AB\end{align*}
- \begin{align*}A'B'\end{align*}
- \begin{align*}A''B''\end{align*}
- Compare the ratios \begin{align*}OA:OA'\end{align*} and \begin{align*}AB:A'B'\end{align*}. What do you notice? Why do you think that is?
- Compare the ratios \begin{align*}OA:OA''\end{align*} and \begin{align*}AB:A''B''\end{align*}. What do you notice? Why do you think that is?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.12.