9-2 Similarity Transformations

Categorize

Options (10)

Preserves Angle Measure

Preserves Orientation vertices

preserves length

preserves angle measure

preserves legnth

preserves angle measur

preserves orientation of vertices

preserves lengt

preserves angle measu

preserves orientation of vertice

Which characteristics applies to each transformation (Match the corresponding cards)

Dilation

Reflection

Rotation

Translation

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Topic 9 Lesson 2

Similarity Transformations

I CAN…
determine whether figures are similar.

VOCABULARY
• similarity transformation

Open Ended

Answer the questions below:

A.) Use Appropriate Tools Is there a composition of transformations that maps ABCD to
the second figure in each student’s work? If so, what is it?

B.) For each student whose work shows a composition of transformations, describe the
relationship between the figures.

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Topic 9 Lesson 2

Graph a Composition of a Rigid Motion and a Dilation

Try It!

1.

The vertices of △XYZ are X(3, 5), Y(−1, 4), and Z(1, 7).

a.

What is the graph of the image (D2∘T〈1, −2〉) (△XYZ)?

b.

What is the graph of the image (D3∘r(90°, O)) (△XYZ)?

EXAMPLE 1

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Topic 9 Lesson 2

Describe a Composition of a Rigid Motion and a Dilation

Is there a composition of transformations that maps △XYZ to △JKL? Explain.

EXAMPLE 2

MAKE SENSE AND PERSEVERE
Think about how you could use reflection, translation, or rotation to create an image of △XYZ with the orientation
opposite of △JKL. Which rigid motion would you use?

Notice that Y in △XYZ is in the upper left of the first quadrant, but its corresponding vertex K in △JKL
is in the lower right of the third quadrant, so it appears that △XYZ is rotated. Since △JKL is larger than
△XYZ , it is also dilated.

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Topic 9 Lesson 2

Describe a Composition of a Rigid Motion and a Dilation

Is there a composition of transformations that maps △XYZ to △JKL? Explain.

So, the composition of transformations D2∘r(180°, O) maps △XYZ to △JKL.

EXAMPLE 2

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Topic 9 Lesson 2

Describe a Composition of a Rigid Motion and a Dilation

Try It!

2.

If the transformations in Example 2 are performed in the reverse order, are the results
the same? Do you think your answer holds for all compositions of transformations?
Explain.

EXAMPLE 2

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Topic 9 Lesson 2

Find Similarity Transformations

Why is PQRS similar to GKJH?

A similarity transformation is a composition of one or more rigid motions and a dilation. A similarity
transformation results in an image that is similar to the preimage.

Measure the angles of the figures to determine that ∠S corresponds to ∠H and ∠R corresponds to ∠J.
The orientation is reversed in GHJK, so the rigid motion includes a reflection.

EXAMPLE 3
CONCEPTUAL UNDERSTANDING

STUDY TIP
A similarity transformation is more precisely a rigid motion, a dilation, or a composition of a rigid motion and a dilation.
Furthermore, the phrase “one or more rigid motions” is not necessary due to Theorem 3-3.

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Topic 9 Lesson 2

Find Similarity Transformations

Why is PQRS similar to GKJH?

CONCEPTUAL UNDERSTANDING
EXAMPLE 3

A composition of a reflection, a translation, and a dilation maps PQRS to GKJH, so PQRS and
GKJH are similar, or PQRS ∼GKJH.

For any figures A, B, and C, the following properties hold.

Reflexive Property of Similarity: A ∼A
Symmetric Property of Similarity: If A ∼B, then B ∼A.
Transitive Property of Similarity: If A ∼B and B ∼C, then A ∼C.

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Topic 9 Lesson 2

Find Similarity Transformations

Try It!

3. Describe a possible similarity transformation for each pair of similar figures shown,
and then write a similarity statement.

a. b.

EXAMPLE 3

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Topic 9 Lesson 2

Determine Similarity

Can the artist copy her sketch to cover an entire wall measuring 15 ft high by 20 ft
wide so her wall mural is similar to her sketch? Explain.

APPLICATION
EXAMPLE 4

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Topic 9 Lesson 2

Determine Similarity

Can the artist copy her sketch to cover an entire wall measuring 15 ft high by 20 ft
wide so her wall mural is similar to her sketch? Explain.

APPLICATION
EXAMPLE 4

MAKE SENSE AND PERSEVERE
Suppose you found the scale factor needed to map the width of the sketch to the width of the wall. Would the results
be the same?

To determine the scale factor, convert the dimensions of the wall into inches.

15 • 12 = 180 20 • 12 = 240

The dimensions of the wall are 180 in. high by 240 in. wide.

Divide the height of the wall by the height of the sketch to determine the scale factor needed to map the
sketch to the height of the wall.

180 ÷ 11 ≈ 16.36

Calculate to see whether the width of the sketch maps to the width of the wall.

16.36 • 14 ≈ 229

Since 229 < 240, the sketch cannot be copied to cover the entire wall.

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Topic 9 Lesson 2

Determine Similarity

Try It!

4.

Suppose the artist cuts 2 inches from the width of her sketch in Example 4. How much
should she cut from the height so she can copy a similar image to cover the wall?

EXAMPLE 4

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Topic 9 Lesson 2

Identify Similar Circles

Write a proof that any two circles are similar.

Given: ⊙P with radius r, ⊙Q with radius s

Prove: ⊙P∼⊙Q

Proof: Translate P to Q, so P′ coincides with Q. Then find a scale factor that dilates ⊙P′
to the circle with radius s.

Let . Then the translation followed by a dilation centered at Q with scale factor k maps
⊙P onto ⊙Q. Since a similarity transformation exists, ⊙P ∼⊙Q.

PROOF
EXAMPLE 5

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Topic 9 Lesson 2

Identify Similar Circles

Try It!

5.

Write a proof that any two squares are similar.

EXAMPLE 5

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Topic 9 Lesson 2

CONCEPT SUMMARY

Similarity

WORDS

•A similarity transformation is a composition of one or more rigid motions and
a dilation.

•Two figures are similar if there is a similarity transformation that maps one to the
other.

•All circles are similar to each other.

DIAGRAMS

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Topic 9 Lesson 2

Do You UNDERSTAND?

1.

ESSENTIAL QUESTION What makes a transformation a similarity transformation?
What is the relationship between a preimage and the image resulting from a similarity
transformation?

2.

Error Analysis Reese described the similarity transformation that maps △ABC to
△XZY. What is Reese’s error?

3.

Vocabulary How are similarity transformations and congruence transformations alike?
How are they different?

4.

Construct Arguments A similarity transformation consisting of a reflection and a
dilation is performed on a figure, and one point maps to itself. Describe one way this
could happen.

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Topic 9 Lesson 2

Do You KNOW HOW?

For Exercises 5 and 6, what are the vertices of each image?

5.

r(90°, O)∘D0.5(ABCD) for A(5, 1), B(−3, 4), C(0, 2), D(4, 6)

6.

(D3∘Rx-axis)(△GHJ) for G(3, 5), H(1, −2), J(−1, 6)

7.

Describe a similarity transformation that maps △SQR to △DEF.

8.

Do the two figures appear to be similar? Use transformations to explain.