Common Core State Standards 9-4 Compositions...

570 Chapter 9 Transformations

9-4 Compositions of IsometriesObjectives To find compositions of isometries, including glide reflections

To classify isometries

In the Solve It, you looked for a way to use two reflections to produce the same image as a given horizontal translation. In this lesson, you will learn that any rigid motion can be expressed as a composition of reflections.

The term isometry means same distance. An isometry is a transformation that preserves distance, or length. So, translations, reflections, and rotations are isometries.

Essential Understanding You can express all isometries as compositions of reflections.

Expressing isometries as compositions of reflections depends on the following theorem.

There are only four kinds of isometries.

You will learn about glide reflections later in the lesson.

Theorem 9-1

The composition of two or more isometries is an isometry.

hsm11gmse_0906_t09573.ai

Translation

Orientations are the same. Orientations are opposite.

Rotation Reflection Glide Reflection

R RRRRR RR

R

Can you find more than one way? Which way is the most efficient?

The blue E is a horizontal translation of the red E. How can you use two reflections, one after the other, to move the red E to the position of the blue E? Copy the figure exactly as shown and draw in the two lines of reflection. Explain how you found the lines.

Lesson Vocabulary

•glidereflection•isometry

LessonVocabulary

G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure . . . Also G-CO.A.5

MP 1, MP 3, MP 6

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 1

Lesson 9-4 CompositionsofIsometries 571

In Lesson 9-1, you learned that a composition of transformations is a combination of two or more transformations, one performed after the other.

Composing Reflections Across Parallel Lines

What is (Rm ∘ RO)( J)? What is the distance of the resulting translation?

As you do the two reflections, keep track of the distance moved by a point P of the preimage.

The red arrow shows the translation. The total distance P moved is 2 # AB. Because <AB

># /, AB is the distance between / and m. The distance of the translation is twice the

distance between / and m.

1. a. Draw parallel lines / and m as in Problem 1. Draw J between / and m. What is the image of (Rm ∘ R/)(J)? What is the distance of the resulting translation?

b. Reasoning Use the results of part (a) and Problem 1. Make a conjecture about the distance of any translation that is the result of a composition of reflections across two parallel lines.

Theorem 9-2 Reflections Across Parallel Lines

A composition of reflections across two parallel lines is a translation.You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″ or Rm(R/(△ABC)) = △A″B″C″.

AA″, BB″, and CC″ are all perpendicular to lines / and m.

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A�

B�

C�

A B

C

A� B�

C��

m


m

�

J


PP�

P�

AB

Step 1 Reflect J across �. PA � AP�, so PP� � 2AP�.

m

�

P moved a total distance of 2AP� � 2P�B, or 2AB.

J J J

Step 2 Reflect the image across m. P�B � BP�, so P�P� � 2P�B.

Got It?

How do you know that PA ∙ AP∙, P∙B ∙ BP ∙, and <AB

> ⊥ O?

All three statements are true by the definition of reflection across a line.

Problem 2


Composing Reflections Across Intersecting Lines

Lines O and m intersect at point C and form a 70∙ angle. What is (Rm ∘ RO)(J)? What are the center of rotation and the angle of rotation for the resulting rotation?

After you do the reflections, follow the path of a point P of the preimage.

J is rotated clockwise about the intersection point of the lines. The center of rotation is C. You know that m∠2 + m∠3 = 70. You can use the definition of reflection to show that m∠1 = m∠2 and m∠3 = m∠4. So, m∠1 + m∠2 + m∠3 + m∠4 = 140. The angle of rotation is 140° clockwise.

2. a. Use the diagram at the right. What is (Rb ∘ Ra)(J)? What are the center and the angle of rotation for the resulting rotation?

b. Reasoning Use the results of part (a) and Problem 2. Make a conjecture about the center of rotation and the angle of rotation for any rotation that is the result of any composition of reflections across two intersecting lines.

Theorem 9-3 Reflections Across Intersecting Lines

A composition of reflections across two intersecting lines is a rotation.

You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″

or Rm(R/(△ABC)) = △A″B″C″.

The figure is rotated about the point where the two lines intersect. In this case, point Q.


A�

B�C�

A

Q

B

m�

CA�

B�

C�


m

C

�

J 70�


m

C

12 3 4

P

P�

P�

�

J

Step 1 Reflect J across �.

Step 2 Reflect the image across m.

Step 3 Draw the angles formed by joining P, P�, and P� to C.

J

J


a

b

C

45�J

Got It?

How do you show that mj1 ∙ mj2?If you draw PP∙ and label its intersection point with line / as A, then PA = P′A andPP′ # /. So, by the Converse of the Angle Bisector Theorem, m∠1 = m∠2.

Any composition of isometries can be represented by either a reflection, translation, rotation, or glide reflection. A glide reflection is the composition of a translation (a glide) and a reflection across a line parallel to the direction of translation. You can map a left paw print onto a right paw print with a glide reflection.

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Problem 3


Finding a Glide Reflection Image

Coordinate Geometry What is (Rx∙0 ∘ T60, ∙57)(△TEX)?

3. Graph △TEX from Problem 3. What is the image of △TEX for the glide reflection (Ry = -2 ∘ T61, 07)(△TEX)?


x

y

O�4 �2

2

1X

ET

First use the translation rule to translate △TEX . Then reflect the translation image of each vertex across the line of reflection.

The image of △TEX for the glide reflection

•Theverticesof△TEX•Thetranslationrule•Thelineofreflection


x

y

O�4

2

2 4X

E

E�

T�

X�

T

Use the translation ruleT<0, �5> (�TEX ) to move �TEX down 5 units.

Reflect the image of �TEX across the line x � 0.

Got It?

Lesson CheckDo you know HOW?Copy the diagrams below. Sketch the image of Z reflected across line a, then across line b.

1. 2.

3. △PQR has vertices P(0, 5), Q(5, 3), and R(3, 1). What are the vertices of the image of △PQR for the glide reflection (Ry = -2 ∘ T63, -17)(△PQR)?

Do you UNDERSTAND? 4. Vocabulary In a glide reflection, what is the relationship

between the direction of the translation and the line of reflection?

5. Error Analysis You reflect △DEF first across line m and then across line n. Your friend says you can get the same result by reflecting △DEF first across line n and then across line m. Explain your friend’s error.


m

n

DF

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Z a

b


Za

Cb65�



Practice and Problem-Solving Exercises

Find the image of each letter after the transformation Rm ∘ RO. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

6. 7. 8.

9. 10. 11.

Graph △PNB and its image after the given transformation.

12. (Ry=3 ∘ T62, 07)(△PNB)

13. (Rx=0 ∘ T60, -37)(△PNB)

14. (Ry=0 ∘ T62, 27)(△PNB)

15. (Ry=x ∘ T6-1, 17)(△PNB)

Use the given points and lines. Graph AB and its image A∙B ∙ after a reflection first across O1 and then across O2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

16. A(1, 5) and B(2, 1); /1: x = 3; /2: x = 7 17. A(2, 4) and B(3, 1); /1: x-axis; /2: y-axis

18. A( -4, -3) and B( -4, 0); /1: y = x; /2: y = -x 19. A(2, -5) and B( -1, -3); /1: y = 0; /2: y = 2

20. A(6, -4) and B(5, 0); /1: x = 6; /2: x = 4 21. A( -1, 0) and B(0, -2); /1: y = -1; /2: y = 1

22. Think About a Plan Let A′ be the point (1, 5). If (Ry=1 ∘ T63, 07)(A) = A′, then what are the coordinates of A?

• How can you work backwards to find the coordinates of A? • Should A be to the left or to the right of A′? • Should A be above or below A′?

PracticeA See Problems 1 and 2.


Fm

�


Mm

�


Tm

�


Lm

85�

C

�


m60�

C

�V


Nm

75�

C

�

See Problem 3.


x

y

O

BN

P

2

2

�2

ApplyB



Describe the isometry that maps the black figure onto the blue figure.

23. 24.

25. Which transformation maps the black triangle onto the blue triangle?

Rx=2 ∘ T60, -37

r(180°, O)

Ry =-12

r(180°, O) ∘ Rx-axis

26. Writing Reflections and glide reflections are odd isometries, while translations and rotations are even isometries. Use what you have learned in this lesson to explain why these categories make sense.

27. Open-Ended Draw △ABC. Describe a reflection, a translation, a rotation, and a glide reflection. Then draw the image of △ABC for each transformation.

28. Reasoning The definition states that a glide reflection is the composition of a translation and a reflection. Explain why these can occur in either order.

Identify each mapping as a translation, reflection, rotation, or glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection.

29. △ABC S △EDC

30. △EDC S △PQM

31. △MNJ S △EDC

32. △HIF S △HGF

33. △PQM S △JLM

34. △MNP S △EDC

35. △JLM S △MNJ

36. △PQM S △KJN

37. △KJN S △ABC

38. △HGF S △KJN


x

y

O1

3

�3

�1�3


x

y

O 1

2

4

�2

�2�4


x

y

O 2

2

�2

�4

�2�4


x

y

O

4

2

1 7

�2

�4

A

B

C

D

EF

G

H

I

J

K N

M P

L Q


39. Describe a glide reflection that maps the black R to the blue R.

40. Reasoning Does an x° rotation about a point P followed by a reflection across a line / give the same image as a reflection across / followed by an x° rotation about P? Explain.

ChallengeC


RR

Apply What You’ve Learned

Look back at the information about the video game on page 543 and review the requirements of Case 1 and Case 2. The graph of the puzzle piece and target area is shown again below.

y

A B

C

F E

D

xO 2�4 �2

�2

�4

4

4

2

In the Apply What You’ve Learned sections in Lessons 9-2 and 9-3, you explored how various translations, reflections, and rotations move the puzzle piece. Now you will think about how to use a composition of transformations to move the puzzle piece to the target area.

a. Explain why any composition of transformations that moves the puzzle piece to the target area must include a reflection.

b. Explain why you must use at least two transformations to move the puzzle piece to the target area.

c. Case 2 does not allow reflections across vertical or horizontal lines. Consider the position of Ry=x (△DEF). What is the relationship of Ry=x (△DEF) to △ABC?

PERFO

RMANCE TA

SK MATHEMATICAL PRACTICESMP 3

Common Core State Standards 9-4 Compositions...

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