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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.

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CK-12 Foundation: Dilation in the Coordinate Plane

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 .

CK-12 Foundation: Dilation in the Coordinate Plane

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}

Answers

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)

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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

dilation

dilation

An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

Image Attributions

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