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7.12: Dilation in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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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? After completing this Concept, you'll be able to solve problems like this one.

Guidance

A dilation makes a figure larger or smaller, but has the same shape as the original. In other words, the dilation is similar to the original. All dilations have a center and a scale factor. The center is the point of reference for the dilation (like the vanishing point in a perspective drawing) and scale factor tells us how much the figure stretches or shrinks. A scale factor is typically labeled k\begin{align*}k\end{align*} and is always greater than zero. Also, if the original figure is labeled ABC\begin{align*}\triangle ABC\end{align*}, for example, the dilation would be ABC\begin{align*}\triangle A'B'C'\end{align*}. The ‘ indicates that it is a copy. This tic mark is said “prime,” so A\begin{align*}A'\end{align*} is read “A prime.” A second dilation would be A′′\begin{align*}A''\end{align*}, read “A double-prime.”

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

If the dilated image is larger than the original, then the scale factor is k>1\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 (x,y)(kx,ky)\begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}. In this Concept, the center of dilation will always be the origin, unless otherwise stated.

Example A

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

The coordinates of ABC\begin{align*}\triangle ABC\end{align*} are A(2,1),B(5,1)\begin{align*}A(2, 1), B(5, 1)\end{align*} and C(3,6)\begin{align*}C(3, 6)\end{align*}. The coordinates of ABC\begin{align*}\triangle A'B'C'\end{align*} are A(6,3),B(15,3)\begin{align*}A'(6, 3), B'(15, 3)\end{align*} and C(9,18)\begin{align*}C'(9, 18)\end{align*}. By looking at the corresponding coordinates, each is three times the original. That means k=3\begin{align*}k = 3\end{align*}.

Again, the center, original point, and dilated point are collinear. Therefore, you can draw a ray from the origin to C,B,\begin{align*}C', B',\end{align*} and A\begin{align*}A'\end{align*} such that the rays pass through C,B,\begin{align*}C, B,\end{align*} and A\begin{align*}A\end{align*}, respectively.

Example B

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

ABCAB=(25)2+(11)2=9=3AC=(23)2+(16)2=26CB=(35)2+(61)2=29ABCAB=(615)2+(33)2=81=9AC=(69)2+(318)2=234=326CB=(915)2+(183)2=261=329\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} = \sqrt{234} = 3 \sqrt{26}\\ & CB = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{29} && C'B' = \sqrt{(9-15)^2 +(18-3)^2} = \sqrt{261} = 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*}.

Example C

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*}

Watch this video for help with the Examples above.

Vocabulary

In the graph above, the blue quadrilateral is the original and the red image is the dilation. A dilation an enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure. Similar figures are the same shape but not necessarily the same size. The center of dilation is the point of reference for the dilation and the scale factor for a dilation tells us how much the figure stretches or shrinks.

Guided Practice

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.

1. \begin{align*}A(3, 9), k = \frac{2}{3}\end{align*}

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

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

Remember that the mapping will be \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}.

1. \begin{align*}A' (2, 6)\end{align*}

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

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

Practice

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.

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

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

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

Multi-Step Problem Questions 6-12 build upon each other.

1. Plot \begin{align*}A(1, 2), B(12, 4), C(10, 10)\end{align*}. Connect to form a triangle.
2. Make the origin the center of dilation. Draw 4 rays from the origin to each point from #6. Then, plot \begin{align*}A'(2, 4), B'(24, 8), C'(20, 20)\end{align*}. What is the scale factor?
3. Use \begin{align*}k =4\end{align*}, to find \begin{align*}A''B''C''\end{align*}. Plot these points.
4. What is the scale factor from \begin{align*}A'B'C'\end{align*} to \begin{align*}A''B''C''\end{align*}?
5. Find (\begin{align*}O\end{align*} is the origin):
1. \begin{align*}OA\end{align*}
2. \begin{align*}AA'\end{align*}
3. \begin{align*}AA''\end{align*}
4. \begin{align*}OA'\end{align*}
5. \begin{align*}OA''\end{align*}
6. Find:
1. \begin{align*}AB\end{align*}
2. \begin{align*}A'B'\end{align*}
3. \begin{align*}A''B''\end{align*}
7. Compare the ratios:
1. \begin{align*}OA:OA'\end{align*} and \begin{align*}AB:A'B'\end{align*}
2. \begin{align*}OA:OA''\end{align*} and \begin{align*}AB:A''B''\end{align*}

For questions 13-18, use quadrilateral \begin{align*}ABCD\end{align*} with \begin{align*}A(1, 5), B(2, 6), C(3, 3)\end{align*} and \begin{align*}D(1, 3)\end{align*} and its transformation \begin{align*}A'B'C'D'\end{align*} with \begin{align*}A'(-3, 1), B'(0, 4), C'(3, -5)\end{align*} and \begin{align*}D'(-3, -5)\end{align*}.

1. Plot the two quadrilaterals in the coordinate plane.
2. Find the equation of \begin{align*}\overleftrightarrow{CC'}\end{align*}.
3. Find the equation of \begin{align*}\overleftrightarrow{DD'}\end{align*}.
4. Find the intersection of these two lines algebraically or graphically.
5. What is the significance of this point?
6. What is the scale factor of the dilation?

Vocabulary Language: English

Dilation

Dilation

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

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

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

Scale Factor

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

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