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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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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

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.

#### Example A

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.

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

For this dilation, the mapping will be $(x, y) \rightarrow (1.5x, 1.5y)$ .

$& E(-4, -2) \rightarrow (1.5(-4), 1.5(-2)) \rightarrow E'(-6, -3)\\& F(1, 4) \rightarrow (1.5(1), 1.5(4)) \rightarrow F'(1.5, 6)\\& G(6, 2) \rightarrow (1.5(6), 1.5(2)) \rightarrow G'(9,3)\\& H(0, -4) \rightarrow (1.5(0),1.5(-4)) \rightarrow H'(0, -6)$

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

#### Example B

Determine the coordinates of $\triangle ABC$ and $\triangle A' B' C'$ and find the scale factor.

The coordinates of the vertices of $\triangle ABC$ are $A(2, 1)$ , $B(5, 1)$ and $C(3, 6)$ . The coordinates of the vertices of $\triangle A'B'C'$ 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 C

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

$& \underline{\triangle ABC} && \underline{\triangle A'B'C'}\\& AB = \sqrt{(2-5)^2 + (1-1)^2} = \sqrt{9} =3 && A'B' = \sqrt{(6-15)^2 + (3-3)^2} = \sqrt{81} = 9\\& AC = \sqrt{(2-3)^2 + (1-6)^2} = \sqrt{26} && A'C' = \sqrt{(6-9)^2+(3-18)^2} = \sqrt{234} = 3 \sqrt{26}\\& CB = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{29} && C'B' = \sqrt{(9-15)^2 +(18-3)^2} = \sqrt{261} = 3 \sqrt{29}$

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

### Guided Practice

Given $A$ and the scale factor, determine the coordinates of the dilated point, $A'$ . You may assume the center of dilation is the origin.

1. $A(3, 9), k = \frac{2}{3}$

2. $A(-4, 6), k = 2$

3. $A(9, -13), k = \frac{1}{2}$

Remember that the mapping will be $(x, y) \rightarrow (kx, ky)$ .

1. $A' (2, 6)$

2. $A'(-8, 12)$

3. $A'(4.5, -6.5)$

### Practice

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. $A(12, 8), B(-4, -16), C(0, 10); k = \frac{3}{4}$

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 Language: English Spanish

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.