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9. Similarity
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Grades 9-12

9.02 Similarity transformations

Lesson
Practice

Introduction

We saw similarity transformations performed with sequences of transformations in 8th grade. That concept is used in this lesson where we will determine whether figures are similar and perform similarity transformations including sequences of rigid transformations and sequences that include transformations from lesson  7.01 Dilations  .

Similarity transformations

Exploration

Drag the blue points on the pre-image to explore how its image changes.

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  1. What sequence of transformations could map the pre-image to the image?
  2. Does the sequence of transformations preserve the angles of the pre-image, the side lengths of the pre-image, or both the angles and the side lengths? How do you know?

When a sequence of transformations is made up of rigid transformations, we can say that the image is congruent and therefore similar to its pre-image. The side lengths and angle measures of the pre-image are preserved.

Two figures are said to be similar if there exists a similarity transformation which maps the pre-image to the image.

Similar

Two figures are similar if their corresponding angles are congruent and their corresponding sides are proportional

Similarity transformation

A series of one or more transformations which results in the image being similar to the pre-image

Rotations, reflections, and translations all result in an image congruent to the pre-image. Since all congruent figures can be considered similar with a ratio of 1:1 (that is, with a scale factor of k=1), these are all similarity transformations as well.

As a dilation enlarges or reduces a shape, the image and pre-image's corresponding angles will be congruent, and the corresponding sides will be proportional. This means that dilations are also similarity transformations. Any combination of these four transformations will maintain similarity.

Examples

Example 1

Consider the figure shown on the graph below. It can be used as an example to show what happens to the segment lengths and angle measures after different types of transformations.

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a

Translate ABCD 3 units to the right, and then find the segment lengths and angle measures for both the pre-image and its image.

Worked Solution
Apply the idea

After we translate ABCD 3 units to the right we get this image:

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We know that two of the side lengths of ABCD and A'B'C'D' are horizontal because the y-coordinates of A and D are both zero, and the y-coordinates of A' and D' are both zero. The y-coordinates of B and C are both 2, and the y-coordinates of B' and C' are both 2.

We know that two of the side lengths of ABCD and A'B'C'D' are vertical because the x-coordinates of A and B are both zero, and the x-coordinates of A' and B' are both 3. The x-coordinates of C and D are both 2, and the x-coordinates of C' and D' are both 5.

This means the segments are perpendicular and all angles on both figures are 90 \degree.

In order to find the segment lengths of the horizontal sides, \overline{AD} and \overline{BC}, we only need to subtract the x-coordinates. 2-0 =2 \text{ units}

In order to find the segment lengths of the horizontal sides, \overline{A'D'} and \overline{B'C'}, we only need to subtract the x-coordinates. 5-3 =2 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{BA} and \overline{CD}, we only need to subtract the y-coordinates. 2-0 =2 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{B'A'} and \overline{C'D'}, we only need to subtract the y-coordinates. 2-0 =2 \text{ units}

Reflect and check

A translation to the right 3 units has a transformation mapping: (x, y) \to (x+3, y)

b

Perform the dilation D_{2,(4,3)}{(ABCD)}. Find the segment and angle lengths for the image.

Worked Solution
Create a strategy

The function notation D_{2,(4,3)}{(ABCD)} specifies to dilate ABCD by a scale factor of 2 about the point (4,3).

Apply the idea
  1. Determine the translations from the center of dilation to the vertices of the figure:
    • \left(4,3\right)\to\left(0,0\right): translate 4 units left and 3 units down
    • \left(4,3\right)\to\left(0,2\right): translate 4 units left and 1 unit down
    • \left(4,3\right)\to\left(2,2\right): translate 2 units left and 1 unit down
    • \left(4,3\right)\to\left(2,0\right): translate 2 units left and 3 units down
  2. Multiply these translations by the scale factor, 2, to find the images of these points under the dilation in relation to the center of dilation:
    • Translate 8 units left and 6 units down
    • Translate 8 unit left and 2 units down
    • Translate 4 units left and 2 units down
    • Translate 4 unit left and 6 units down
  3. Apply these translations to the center of dilation, \left(4,3\right), to determine the coordinates of the images of the vertices under dilation:
    • Translating \left(4,3\right) by 8 units left and 6 units down gives us \left(-4,-3\right)
    • Translating \left(4,3\right) by 8 units left and 2 units down gives us \left(-4,1\right)
    • Translating \left(4,3\right) by 4 units left and 2 units down gives us \left(0,1\right)
    • Translating \left(4,3\right) by 4 units left and 6 units down gives us \left(0,-3\right)
  4. Plot the image of the dilation, using the images of the vertices determined in the previous step.
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We know that two of the side lengths of LMNP are horizontal because the y-coordinates of L and P are both -3. The y-coordinates of M and N are both 1.

We know that two of the side lengths of LMNP are vertical because the x-coordinates of L and M are both -4. The x-coordinates of N and P are both 0.

This means the segments are perpendicular and all angles are 90 \degree.

In order to find the segment lengths of the horizontal sides, \overline{LP} and \overline{MN}, we only need to subtract the x-coordinates. 0-(-4) =4 \text{ units}

In order to find the segment lengths of the vertical sides, \overline{ML} and \overline{NP}, we only need to subtract the y-coordinates. 1-(-3) =4 \text{ units}

c

State the relationships between the corresponding sides and angles of the pre-image and image after the translation in part (a). Then state the relationships between the corresponding pre-image and image after the dilation in part (b).

Worked Solution
Create a strategy

Identify whether the transformations preserved distance, angles, or distance and angles.

Apply the idea

  • We can see that distance has been preserved when translating ABCD since the side lengths of the image are the same as the pre-image in part (a). The angle measurements are all 90 \degree. Therefore, a translation to the right 3 units preserves both distance and angle measurements.

  • We can see that distance has not been preserved when dilating ABCD since the side lengths of the image are twice as large as the pre-image in part (b). The angle measurements are all 90 \degree. Therefore, a dilation using a scale factor of 2 preserves only angle measurements.

Example 2

The following sequences of transformations are applied to figure ABCD. Without performing the transformations, determine whether each final image will be a similarity transformation, a congruency transformation, or both. Explain your reasoning.

a

A reflection across the x-axis followed by a rotation 90 \degree clockwise about the origin.

Worked Solution
Create a strategy

A reflection across the x-axis has a transformation mapping: (x,y) \to (x, -y)

A rotation 90 \degree clockwise about the origin has a transformation mapping: (x, y) \to (y, -x)

Apply the idea

Since a reflection across the x-axis has a transformation mapping (x,y) \to (x, -y), we know that the coordinates were not multiplied by a scale factor, meaning no dilation occurred.

After this rigid transformation, a rotation 90 \degree clockwise about the origin with a transformation mapping (x, y) \to (y, -x) also does not change the segments of the figure, since there is no scale factor other than 1.

The sequence of rigid transformations will change the orientation of ABCD, while preserving its angles and side lengths. Since the sequence of transformations will lead to a congruent image to the pre-image, the sequence is both a congruency transformation and a similarity transformation.

b

A dilation by a scale factor of \dfrac{1}{3} with the center of dilation at the origin followed by a rotation 180 \degree.

Worked Solution
Apply the idea

Since the scale factor k <1, any segment on the image will be k times the size of the line segments of the pre-image, so the side lengths will be proportional, but not congruent, while the angles will be preserved.

A rotation will preserve both the side lengths and angle measures of the figure, after it has been dilated.

The sequence of transformations will be a similarity transformation.

Reflect and check

The coordinate mapping for the sequence of transformations is (x,y) \to (-\dfrac{1}{3}x, -\dfrac{1}{3}y)

Multiplying the x- and y-coordinates by the same scale factor indicates a dilation, which creates a similar, but not congruent, transformation.

Idea summary

  • Translations, reflections, and rotations are rigid transformations that preserve both angle measures and side lengths. Transformation sequences containing only rigid transformations are both a similarity transformation and a congruency transformation.

  • Dilations preserve angle measures and lead to proportional side lengths between an image and its pre-image. Therefore, any transformation sequence that includes a dilation will be a similarity transformation, but not a congruency transformation.

Similarity ratios and statements

When two figures are similar, we express this using a similarity statement and can identify the similarity ratio of each pair of corresponding sides.

Similarity ratio

The ratio of two corresponding side lengths in a pair of similar figures.

Similarity statement

A statement that indicates two polygons are similar by listing the vertices in the order of the correspondence.

Triangles A B C and D E F. Angles B and E are congruent.

The similarity statement for the triangles in the diagram shown is \triangle ABC \sim \triangle DEF.

Similarity statements can be written in any order which keeps corresponding vertices in order. For this example we could have written the similarity statement in several different ways, though it is most common to arrange the vertices in alphabetical order (when possible).

If two figures are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

Examples

Example 3

Determine if the two triangles are similar or not. If so, write a similarity statement.

Triangles C D E and L M N. C D has length 7, D E has length 4, and E C has length 5. L M has length 42, M N has length 24, and N L has length 30.
Worked Solution
Create a strategy

Determine which sides are corresponding and then check that each pair of corresponding sides are in the same proportion.

Apply the idea

The side lengths of the smaller triangle in ascending order are: 4,5,7.

The side lengths of the larger triangle in ascending order are: 24,30,42.

Doing this tells us that the corresponding side pairs are:

  • 4 and 24
  • 5 and 30
  • 7 and 42

To test for similarity, we want to check whether the corresponding pairs of sides are all in the same proportion.

\frac{24}{4}=6, \quad \frac{30}{5}=6, \quad\frac{42}{7}=6

Since all the pairs of corresponding sides are in the same proportion, these triangles are similar.

The similarity statement is CDE \sim LMN.

Reflect and check

In order to justify that two triangles are similar, we will need to check all three corresponding sides. However, to show that two triangles are not similar, we only have to find two corresponding side pairs which are not in the same proportion.

Example 4

Identify the coordinates of the figure shown after a rotation by 180 \degree about the origin and a dilation by a similarity ratio of \dfrac{1}{2} with the origin as the center of dilation.

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Worked Solution
Create a strategy

A rotation 180 \degree counterclockwise has a transformation mapping: \left(x,y \right) \to \left(-x,-y\right)

A dilation by a scale factor of \dfrac{1}{2} has a transformation mapping: \left(x,y\right) \to \left(\dfrac{1}{2}x,\dfrac{1}{2}y\right)

Apply the idea

Applying these transformations in order to the vertices of the pre-image, we get:

A(-4,3)A'(4, -3)L(2, -1.5)
B(-1,2)B'(1,-2)M(0.5, -1)
C(-3,-1)C'(3,1)N(1.5, 0.5)
Reflect and check

We can combine the two transformations into one mapping: \left(x,y \right) \to \left(-\dfrac{1}{2}x,-\dfrac{1}{2}y\right)

The image of ABC after the sequence of similarity transformations is graphed:

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Example 5

Use similarity transformations to determine whether or not the two figures are similar. If they are similar, write a similarity statement and describe the required transformations. If not, explain how you know they aren't similar.

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A(3,-1)A'(-6, -2)
B(1,-8)B'(-2,-16)
C(5,-8)C'(-10,-16)
D(7,-1)D'(-14,-2)
Worked Solution
Create a strategy

Start by looking at the graphs of the overall figures and visualizing how the transformation could have happened. Since the image is oriented differently from the pre-image, we may consider a reflection or a rotation. The image seems to be a reflection across the y-axis.

Since the image is an enlargement of the pre-image, we know that the scale factor k>1. This means the pre-image was dilated. In order to determine the scale factor, we can find the length of corresponding side lengths on each figure and calculate the scale factor.

AD is 4 units and A'D' is 8 units. The side lengths of the pre-image would be multiplied by 2 to lead to the side lengths of the image, so that scale facor must be 2.

Apply the idea

With closer inspection of the coordinates, we can see that the x-coordinates of ABCD changed signs, suggesting a reflection across the y-axis. Reflecting ABCD across the y-axis would lead to the coordinates (-3,-1), (-1, -8), (-5, -8), (-7,-1).

If we test out multiplying these coordinates by 2 with a center of dilation at (0,0), the result will be the coordinates of A'B'C'D', meaning that following a reflection across the y-axis, the figure was dilated by a scale factor of 2.

If we consider AD=4 and A'D'=8, we get the following similarity ratio:\frac{A'D'}{AD}=\frac{8}{4}=2

Since these transformations lead to A'B'C'D' being similar to ABCD, we can write the similarity statement ABCD \sim A'B'C'D'.

The similarity transformation from ABCD to A'B'C'D' can be described as "Reflect the pre-image over the y-axis, then dilate around the origin by a scale factor of 2."

Reflect and check

A reflection across the y-axis has a transformation mapping: \left(x,y\right) \to \left(-x,y\right)

A dilation by a scale factor of 2 has a transformation mapping: \left(x,y\right) \to \left(2x,2y\right)

Applying these transformations in order to one of the vertices A\left(3,-1\right) we get:\left(3,-1\right) \to \left(-3,-1\right)

and then \left(-3,-1\right) \to \left(-6,-2\right)

In some instances there are multiple correct similarity transformations which map the pre-image on to the image. For this particular similarity transformation, the dilation could also have occurred before the reflection over the y-axis.

Idea summary

Consider two ways that we can determine the similarity between two figures:

  • By showing that all the side lengths from the pre-image to the image are proportional
  • By showing that a sequence of similarity transformations will map the pre-image to the image

Outcomes

G.CO.A.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

G.SRT.A.2

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

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