# 7.6: Similarity Transformations

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Draw a dilation of a given figure.
• Plot an image when given the center of dilation and scale factor.
• Determine if one figure is the dilation of another.

## Review Queue

1. Are the two quadrilaterals similar? How do you know?
2. What is the scale factor from \begin{align*}XYZW\end{align*} to \begin{align*}CDAB\end{align*}? Leave as a fraction.

Know What? One practical application of dilations is perspective drawings. These drawings use a vanishing point (the point where the road meets the horizon) to trick the eye into thinking the picture is three-dimensional. The picture to the right is called a one-point perspective. They are typically used to draw streets, train tracks, or anything that is linear.

Your task for this Know What? is to draw your own perspective drawing with one vanishing point and at least 4 objects (buildings, cars, sidewalk, train tracks, etc).

## Dilations

A dilation makes a figure larger or smaller and has the same shape as the original.

Dilation: An enlargement or reduction of a figure that preserves size but not shape. 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 scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled \begin{align*}k\end{align*} and always greater than zero. A dilation, or copy, is always followed by \begin{align*}a\end{align*} ‘.

Label It Say It
“prime” (copy of the original)
\begin{align*}A'\end{align*} “a prime” (copy of point \begin{align*}A\end{align*})
\begin{align*}A''\end{align*} “a double prime” (second copy)

Example 1: The center of dilation is \begin{align*}P\end{align*} and the scale factor is 3.

Find \begin{align*}Q'\end{align*}.

Solution: If the scale factor is 3 and \begin{align*}Q\end{align*} is 6 units away from \begin{align*}P\end{align*}, then \begin{align*}Q'\end{align*} is going to be \begin{align*}6 \times 3 = 18\end{align*} units away from \begin{align*}P\end{align*}. The dilation will be on the same line as the original and center.

Example 2: Using the picture above, change the scale factor to \begin{align*}\frac{1}{3}\end{align*}.

Find \begin{align*}Q''\end{align*}.

Solution: The scale factor is \begin{align*}\frac{1}{3}\end{align*}, so \begin{align*}Q''\end{align*} is going to be \begin{align*}6 \times \frac{1}{3} = 2\end{align*} units away from \begin{align*}P\end{align*}. \begin{align*}Q''\end{align*} will also be collinear with \begin{align*}Q\end{align*} and center.

Example 3: \begin{align*}KLMN\end{align*} is a rectangle. If the center of dilation is \begin{align*}K\end{align*} and \begin{align*}k = 2\end{align*}, draw \begin{align*}K'L'M'N'\end{align*}.

Solution: If \begin{align*}K\end{align*} is the center of dilation, then \begin{align*}K\end{align*} and \begin{align*}K'\end{align*} will be the same point. From there, \begin{align*}L'\end{align*} will be \begin{align*}8\end{align*} units above \begin{align*}L\end{align*} and \begin{align*}N'\end{align*} will be 12 units to the right of \begin{align*}N\end{align*}.

Example 4: Find the perimeters of \begin{align*}KLMN\end{align*} and \begin{align*}K'L'M'N'\end{align*}. Compare this ratio to the scale factor.

Solution: The perimeter of \begin{align*}KLMN = 12 + 8 + 12 + 8 = 40\end{align*}. The perimeter of \begin{align*}K'L'M'N' = 24 + 16 + 24 + 16 = 80\end{align*}. The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.

Example 5: \begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}. If \begin{align*}P\end{align*} is the center of dilation, what is the scale factor?

Solution: Because \begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}, then \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}. The scale factor is the ratio of the sides. Since \begin{align*}\triangle ABC\end{align*} is smaller than the original, \begin{align*}\triangle DEF\end{align*}, the scale factor is going to be less than one, \begin{align*}\frac{12}{20} = \frac{3}{5}\end{align*}.

If \begin{align*}\triangle DEF\end{align*} was the dilated image, the scale factor would have been \begin{align*}\frac{5}{3}\end{align*}.

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

If the dilated image is larger than the original, then \begin{align*}k>1\end{align*}.

## Dilations in the Coordinate Plane

In this text, the center of dilation will always be the origin.

Example 6: 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.

Solution: 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 \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}.

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*}

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

Example 7: Determine the coordinates of \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle A' B' C'\end{align*} and find the scale factor.

Solution: The coordinates of \begin{align*}\triangle ABC\end{align*} are \begin{align*}A(2, 1)\end{align*}, \begin{align*}B(5, 1)\end{align*} and \begin{align*}C(3, 6)\end{align*}. The coordinates of \begin{align*}\triangle A'B'C'\end{align*} are \begin{align*}A'(6, 3), B'(15, 3)\end{align*} and \begin{align*}C'(9, 18)\end{align*}. Each of the corresponding coordinates are three times the original, so \begin{align*}k = 3\end{align*}.

Example 8: Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example 7.

Solution:

\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*}.

Know What? Revisited Answers to this project will vary depending on what you decide to draw. Make sure that you have at least four objectsrt of detail. If you are having trouble getting started, go to the website: http://www.drawing-and-painting-techniques.com/drawing-perspective.html

## Review Questions

• Questions 1-6 are similar to Examples 1 and 2.
• Questions 7-10 are similar to Example 3.
• Questions 11-18 are similar to Example 5.
• Questions 19-24 are similar to Examples 6 and 7.
• Questions 25-30 are similar to Example 8.

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*}

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*}

For the given shapes, draw the dilation, given the scale factor and center.

1. \begin{align*}k=3.5\end{align*}, center is \begin{align*}A\end{align*}

1. \begin{align*}k=2\end{align*}, center is \begin{align*}D\end{align*}

1. \begin{align*}k = \frac{3}{4}\end{align*}, center is \begin{align*}A\end{align*}

1. \begin{align*}k = \frac{2}{5}\end{align*}, center is \begin{align*}A\end{align*}

In the four questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black figure is the original and \begin{align*}P\end{align*} is the center of dilation.

1. \begin{align*}k = 4\end{align*}

1. \begin{align*}k = \frac{1}{3}\end{align*}

1. \begin{align*}k = 2.5\end{align*}

1. \begin{align*}k = \frac{1}{4}\end{align*}

In the four questions below, find the scale factor, given the corresponding sides. In each diagram, the black figure is the original and \begin{align*}P\end{align*} is the center of dilation.

The origin is the center of dilation. Draw 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 21-24 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 #21. 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*}?

If \begin{align*}O\end{align*} is the origin, find the following lengths (using 21-24 above). Round all answers to the nearest hundredth.

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. \begin{align*}AB\end{align*}
7. \begin{align*}A'B'\end{align*}
8. \begin{align*}A''B''\end{align*}
9. Compare the ratios \begin{align*}OA:OA'\end{align*} and \begin{align*}AB:A'B'\end{align*}. What do you notice? Why do you think that is?
10. Compare the ratios \begin{align*}OA:OA''\end{align*} and \begin{align*}AB:A''B''\end{align*}. What do you notice? Why do you think that is?

1. Yes, all the angles are congruent and the corresponding sides are in the same ratio.
2. \begin{align*}\frac{5}{3}\end{align*}
3. Yes, \begin{align*}LMNO \sim EFGH\end{align*} because \begin{align*}LMNO\end{align*} is exactly half of \begin{align*}EFGH\end{align*}.

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