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# Dilation in the Coordinate Plane

## Multiplication of coordinates by a scale factor.

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Dilation in the Coordinate Plane

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 < k < 1\end{align*}.

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

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

#### Example C

Quadrilateral EFGH\begin{align*}EFGH\end{align*} has vertices E(4,2),F(1,4),G(6,2)\begin{align*}E(-4, -2), F(1, 4), G(6, 2)\end{align*} and H(0,4)\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 (x,y)(1.5x,1.5y)\begin{align*}(x, y) \rightarrow (1.5x, 1.5y)\end{align*}.

E(4,2)(1.5(4),1.5(2))E(6,3)F(1,4)(1.5(1),1.5(4))F(1.5,6)G(6,2)(1.5(6),1.5(2))G(9,3)H(0,4)(1.5(0),1.5(4))H(0,6)

Watch this video for help with the Examples above.

### Guided Practice

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

1. A(3,9),k=23\begin{align*}A(3, 9), k = \frac{2}{3}\end{align*}

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

3. A(9,13),k=12\begin{align*}A(9, -13), k = \frac{1}{2}\end{align*}

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

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

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

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

### Explore More

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

1. A(8,2),A(12,3)\begin{align*}A(8, 2), A'(12, 3)\end{align*}
2. A(5,9),A(45,81)\begin{align*}A(-5, -9), A'(-45, -81)\end{align*}
3. A(22,7),A(11,3.5)\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. A(2,4),B(3,7),C(1,2);k=3\begin{align*}A(2, 4), B(-3, 7), C(-1, -2); k = 3\end{align*}
2. A(12,8),B(4,16),C(0,10);k=34\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 A(1,2),B(12,4),C(10,10)\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 A(2,4),B(24,8),C(20,20)\begin{align*}A'(2, 4), B'(24, 8), C'(20, 20)\end{align*}. What is the scale factor?
3. Use k=4\begin{align*}k =4\end{align*}, to find A′′B′′C′′\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.