# 7.16: Dilation in the Coordinate Plane

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Multiplication of coordinates by a scale factor given the origin as the center.

## 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 $$k$$. Only positive scale factors, $$k$$, will be considered in this text.

If the dilated image is smaller than the original, then $$0<k<1$$.

If the dilated image is larger than the original, then $$k>1$$.

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 $$(x,y)\rightarrow (kx,ky)$$. 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?

For Examples 1 and 2, use the following instructions:

Given A and the scale factor, determine the coordinates of the dilated point, $$A′$$. You may assume the center of dilation is the origin. Remember that the mapping will be $$(x, y)\rightarrow (kx, ky)$$.

Example $$\PageIndex{1}$$

$$A(−4,6), k=2$$

Solution

$$A′(−8,12)$$

Example $$\PageIndex{2}$$

$$A(9,−13), k=\dfrac{1}{2}$$

Solution

$$A′(4.5,−6.5)$$

Example $$\PageIndex{3}$$

Quadrilateral EFGH\) has vertices $$E(−4,−2)$$, $$F(1,4)$$, $$G(6,2)$$ and $$H(0,−4)$$. Draw the dilation with a scale factor of 1.5. Figure $$\PageIndex{1}$$

Solution

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)\rightarrow (1.5x, 1.5y) . \(\begin{aligned} &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{aligned}

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

Example $$\PageIndex{4}$$

Determine the coordinates of $$\Delta ABC$$ and $$\Delta A′B′C′$$ and find the scale factor. Figure $$\PageIndex{2}$$

Solution

The coordinates of the vertices of $$\Delta ABC$$ are $$A(2,1)$$, B(5,1)\) and C(3,6)\). The coordinates of the vertices of $$\Delta A′B′C′$$ are A′(6,3)\), $$B′(15,3)$$ and C′(9,18)\). Each of the corresponding coordinates are three times the original, so $$k=3$$.

Example $$\PageIndex{5}$$

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

Solution

$$\begin{array}{ll} \underline{\Delta ABC} & \underline{\Delta A'B'C'} A B=\sqrt{(2-5)^{2}+(1-1)^{2}}=\sqrt{9}=3 & A^{\prime} B^{\prime}=\sqrt{(6-15)^{2}+(3-3)^{2}}=\sqrt{81}=9 \\ A C=\sqrt{(2-3)^{2}+(1-6)^{2}}=\sqrt{26} & A^{\prime} C^{\prime}=\sqrt{(6-9)^{2}+(3-18)^{2}}=3 \sqrt{26} \\ C B=\sqrt{(3-5)^{2}+(6-1)^{2}}=\sqrt{29} & C^{\prime} B^{\prime}=\sqrt{(9-15)^{2}+(18-3)^{2}}=3 \sqrt{29} \end{array}$$

From this, we also see that all the sides of $$\Delta A′B′C′$$ are three times larger than $$\Delta ABC$$.

## Review

Given $$A$$ and $$A′$$, find the scale factor. You may assume the center of dilation is the origin.

1. $$A(8,2), A′(12,3)$$
2. $$A(−5,−9), A′(−45,−81)$$
3. $$A(22,−7), A(11,−3.5)$$

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

1. $$A(2,4), B(−3,7), C(−1,−2); k=3$$
2. $$A(12,8), B(−4,−16), C(0,10); k=34$$

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

1. Plot $$A(1,2), B(12,4), C(10,10)$$. Connect to form a triangle.
2. Make the origin the center of dilation. Draw 4 rays from the origin to each point from #21. Then, plot $$A′(2,4), B′(24,8), C′(20,20)$$. What is the scale factor?
3. Use $$k=4$$, to find $$A′′B′′C′′$$. Plot these points.
4. What is the scale factor from $$A′B′C′$$ to $$A′′B′′C′′$$?

If $$O$$ is the origin, find the following lengths (using 6-9 above). Round all answers to the nearest hundredth.

1. $$OA$$
2. $$AA′$$
3. $$AA′′$$
4. $$OA′$$
5. $$OA′′$$
6. $$AB$$
7. $$A′B′$$
8. $$A′′B′′$$
9. Compare the ratios $$OA:OA′$$ and $$AB: A′B′$$. What do you notice? Why do you think that is?
10. Compare the ratios $$OA:OA′′$$ and $$AB: A′′B′′$$. What do you notice? Why do you think that is?

## Vocabulary

Term Definition
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.

Interactive Element

Video: Dilation in the Coordinate Plane Principles - Basic

Activities: Dilation in the Coordinate Plane Discussion Questions

Study Aids: Types of Transformations Study Guide

Practice: Dilation in the Coordinate Plane

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