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

-15

-10

-5

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).

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.

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.

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)

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

Applying 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)

Example 3

Determine if the two triangles are similar or not.

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.

Approach

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

Solution

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.

Reflection

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.

Outcomes

M2.G.CO.A.1

Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.

M2.G.SRT.A.1

Use properties of dilations given by a center and a scale factor to solve problems and to justify relationships in geometric figures.

M2.G.SRT.A.2

Define similarity in terms of transformations. Use transformations to determine whether two figures are similar.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

9.02 Similarity transformations

Concept summary

Worked examples

Example 1

Solution

Reflection

Approach

Solution

Reflection

Approach

Solution

Example 2

Approach

Solution

Approach

Solution

Reflection

Example 3

Approach

Solution

Reflection

Outcomes

M2.G.CO.A.1

M2.G.SRT.A.1

M2.G.SRT.A.2

M2.MP1

M2.MP3

M2.MP4

M2.MP5

M2.MP6

M2.MP7

M2.MP8

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