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7. Similarity
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United States of AmericaFL
Grade 912

7.02 Similarity transformations

Lesson
Practice
Lesson

Concept summary

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.

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.

Worked examples

Example 1

-15
-10
-5
5
10
15
x
-15
-10
-5
5
10
15
y
a

Write a similarity statement for the above pre-image and image.

Solution

ABCD \sim A'B'C'D'

Reflection

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).

b

Describe the similarity transformation from ABCD to A'B'C'D'.

Approach

A similarity transformation written as simply as possible is a sequence of transformations where one is a dilation and any others are distinct rigid transformations.

It can be helpful to focus on transforming a specific point or segment on the pre-image and then see if the transformation applies to entire pre-image. For example, we see that A(3,-1) \to A'(-6,-2), which provides information about the dilation and other transformation that take place.

Solution

Since each coordinate was multiplied by 2 and the x-coordinate changed signs, this suggests that there was a reflection over the y-axis and a dilation with a scale factor of 2. We can check the other vertex pairs to confirm this.

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

Reflection

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.

c

Determine the similarity ratio of the two figures.

Approach

We can compare any two corresponding side lengths to determine the similarity ratio. The similarity ratio is always the length of the side in the image divided by the length of the side in the preimage.

Solution

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

Example 2

Identify the coordinates of \left(-9,2\right) after each sequence of transformations.

a

Dilate by a scale factor of \dfrac{1}{3}

Approach

The dilation \left(x,y\right) \to \left(kx,ky\right) takes the pre-image and dilates it by a factor of k. In this case we have a scale factor of \dfrac{1}{3}.

Solution

The transformation mapping is \left(x,y\right) \to \left(\dfrac{1}{3}x,\dfrac{1}{3}y\right).

Applying this to the point \left(-9,2\right) we get:\left(-9,2\right) \to \left(-3,\dfrac{2}{3}\right)

b

Rotate by 180 \degree about the origin and then dilate by a scale factor of 4

Approach

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 4 has a transformation mapping: \left(x,y\right) \to \left(4x,4y\right)

Solution

Aplying these transformations in order to the point \left(-9,2\right) we get: \left(-9,2\right) \to \left(9, -2\right) and then\left(9, -2\right) \to \left(36, -8\right)

Reflection

We can combine the two transformations into one mapping: \left(x,y \right) \to \left(-4x,-4y\right)

Outcomes

MA.912.GR.2.1

Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

MA.912.GR.2.2

Identify transformations that do or do not preserve distance.

MA.912.GR.2.3

Identify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

MA.912.GR.2.5

Given a geometric figure and a sequence of transformations, draw the transformed figure on a coordinate plane.

MA.912.GR.2.8

Apply an appropriate transformation to map one figure onto another to justify that the two figures are similar.

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