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7. Similarity
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Geometry

7.02 Similarity transformations

Lesson
Practice

Similarity transformations

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.

Exploration

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

Loading interactive...
  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

The following sequences of transformations are applied to figure ABCD. Without performing the transformations, determine whether each final image will be similar, congruent, 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 coordinate 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 coordinate 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 image.

Example 2

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

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Worked Solution
Create a strategy

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

A dilation by a scale factor of \dfrac{1}{2} has a coordinate 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:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Example 3

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.

-15
-10
-5
5
10
15
x
-15
-10
-5
5
10
15
y
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 coordinate mapping: \left(x,y\right) \to \left(-x,y\right)

A dilation by a scale factor of 2 has a coordinate 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

  • 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.

Outcomes

G.RLT.3ci

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: i) translations;

G.RLT.3cii

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: ii) reflections over any horizontal or vertical line or the lines y = x or y = -x;

G.RLT.3ciii

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: iii) clockwise or counterclockwise rotations of 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation is limited to the origin;

G.RLT.3civ

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: iv) dilations, from a fixed point on a coordinate grid.

G.TR.3

The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles.

G.TR.3d

Describe a sequence of transformations that can be used to verify similarity of triangles located in the same plane.

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