content standards 9-4 compositions of g.co.5 . . . specify

570 Chapter 9 Transformations

9-4 Compositions of IsometriesObjectives To �nd compositions of isometries, including glide re�ections

To classify isometries

In the Solve It, you looked for a way to use two re�ections 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 re�ections.

�e term isometry means same distance. An isometry is a transformation that preserves distance, or length. So, translations, re�ections, and rotations are isometries.

Essential Understanding You can express all isometries as compositions of re�ections.

Expressing isometries as compositions of re�ections depends on the following theorem.

�ere are only four kinds of isometries.

You will learn about glide re�ections later in the lesson.

Theorem 9-1

�e 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

isometry

LessonVocabularyLesson VocabularyLessonLesson LessonLesson VocabularyVocabularyVocabularyVocabularyVocabulary

Content StandardsG.CO.5 . . . Specify a sequence of transformation that will carry a given figure onto another.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure . . .

MATHEMATICAL PRACTICES

GEOM12_SE_CCS_C09L04.indd 570 6/30/11 5:12:03 PM

1 Common Core

CC-20CC-13

HSM12GESE_CC13.indd 1 7/28/11 5:34:01 PM

Problem 1

Got It?

Lesson 9-4 571

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 R<)( J)? What is the distance of the resulting translation?

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

�e red arrow shows the translation. �e total distance P moved is 2 AB. Because AB , AB is the distance between and m. �e 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 re�ections across two parallel lines.

Theorem 9-2 Reflections Across Parallel Lines

A composition of re�ections 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.

geom12_se_ccs_c09l04_t01.ai

A

B

C

A B

C

A B

Cm


m

J


PP

P

AB

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

m

PP 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.

How do you know that PA AP , P B BP , and AB <?All three statements are true by the definition of reflection across a line.


CC-13 Compositions of Isometries 2


Problem 2

Got It?


Composing Reflections Across Intersecting Lines

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

After you do the re�ections, follow the path of a point P of the preimage.

J is rotated clockwise about the intersection point of the lines. �e center of rotation is C. You know that m 2 m 3 70. You can use the de�nition of re�ection to show that m 1 m 2 and m 3 m 4. So, m 1 m 2 m 3 m 4 140. �e 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 re�ections across two intersecting lines.

Theorem 9-3 Reflections Across Intersecting Lines

A composition of re�ections 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 .

�e �gure is rotated about the point where the two lines intersect. In this case, point Q.


A

BC

A

Q

B

m

CA

B

C


m

C

J 70


m

C

12 3 4

P

P

PJ

Step 1 Reflect J across .

Step 2 Reflect the image across m.

P

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

J

J


a

b

C

45J

J is rotated clockwise about the intersection point of the lines. �e center of rotation is show that �e angle of rotation is

How do you show that m 1 m 2?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 re�ection, translation, rotation, or glide re�ection. A glide reflection is the composition of a translation (a glide) and a re�ection across a line parallel to the direction of translation. You can map a left paw print onto a right paw print with a glide re�ection.

hsm11gmse_0906_t14038


3 Common Core


Problem 3

Got It?

Lesson 9-4 573

Finding a Glide Reflection Image

Coordinate Geometry What is (Rx 0 T 0, 5 )( TEX)?

3. Graph TEX from Problem 3. What is the image of TEX for the glide re�ection (Ry 2 T 1, 0 )( TEX)?


x

y

O4 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.

First use the translation rule to translate translation image of each vertex across the line of reflection.

The image of TEX for the glide

reflection

The image of

reflection

TEX


x

y

O4

2

2 4X

E

E

T

X

T

Use the translation ruleT<0, 5> ( TEX ) to move

TEX down 5 units.

x

44

TT

Reflect the image of TEX across the line x 0.

Lesson CheckDo you know HOW?Copy the diagrams below. Sketch the image of Z re�ected 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 re�ection (Ry 2 T 3, 1 )( PQR)?

Do you UNDERSTAND?4. Vocabulary In a glide re�ection, what is the relationship

between the direction of the translation and the line of re�ection?

5. Error Analysis You re�ect DEF �rst across line m and then across line n. Your friend says you can get the same result by re�ecting DEF �rst across line n and then across line m. Explain your friend’s error.



Z a

b


Za

Cb65






Practice and Problem-Solving Exercises

Find the image of each letter after the transformation Rm R<. 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 image after the given transformation.

12. (Ry 3 T 2, 0 )( PNB)

13. (Rx 0 T 0, 3 )( PNB)

14. (Ry 0 T 2, 2 )( PNB)

15. (Ry x T 1, 1 )( PNB)

Use the given points and lines. Graph AB and its image A B after a re�ection �rst across <1 and then across <2. 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 T 3, 0 )(A) A , then what are the coordinates of A?

How can you work backwards to �nd the coordinates of A?Should A be to left or to the right of A ?Should A be above or below A ?

PracticeA See Problems 1 and 2.


Fm


Mm


Tm


Lm

85C


m60

C

V


Nm

75C

See Problem 3.


x

y

O

BN

P

2

2

2

ApplyB



5 Common Core


Lesson 9-6 575

Describe the isometry that maps the black �gure onto the blue �gure.

23. 24.

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

Rx 2 T 0, 3

r(180 , O)

Ry 12

r(180 , O) Rx-axis

26. Writing Re�ections and glide re�ections 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 re�ection, a translation, a rotation, and a glide re�ection. �en draw the image of ABC for each transformation.

28. Reasoning �e de�nition states that a glide re�ection is the composition of a translation and a re�ection. Explain why these can occur in either order.

Identify each mapping as a translation, re�ection, rotation, or glide re�ection. Write the rule for each translation, re�ection, rotation, or glide re�ection. For glide re�ections, write the rule as a composition of a translation and a re�ection.

29. ABC EDC

30. EDC PQM

31. MNJ EDC

32. HIF HGF

33. PQM JLM

34. MNP EDC

35. JLM MNJ

36. PQM KJN

37. KJN ABC

38. HGF KJN


x

y

O1

3

3

13


x

y

O 1

2

4

2

24


x

y

O 2

2

2

4

24


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 re�ection that maps the black R to the blue R.

40. Reasoning Does an x rotation about a point P followed by a re�ection across a line give the same image as a re�ection across followed by an x rotation about P? Explain.

ChallengeC


RR

Mixed Review

Coordinate Geometry Find the image of ABC for the rotation described.

45. A(0, 4), B(0, 0), C( 3, 1); r(90 , O)( ABC) 46. A(4, 2), B(2, 8), C(8, 0); r(180 , O)( ABC)

Identify the two statements that contradict each other.

47. I. ABC is a right triangle.

II. ABC is equiangular.

III. ABC is isosceles.

See Lesson 9-3.

See Lesson 5-5.

48. I. In right ABC, m B 90.

II. In right ABC, m A 80.

III. In right ABC, m C 90.

Get Ready! To prepare for Lesson 9-5, do Exercises 49–51.

Determine whether the triangles must be congruent. If so, name the postulate or theorem that justi�es your answer. If not, explain.

49. 50. 51.

See Lesson 4-2.


BJ

K

L

A C


35

4

3


12520

20

125

Standardized Test Prep

41. What is (Rx 0 T 12, 6 )(11, 5)?

(1, 11) ( 1, 11) (1, 11) ( 1, 11)

42. ABCD is a rectangular window divided into 12 panes of glass. E, F, G, and H are midpoints of AB, BC , CD, and AD, respectively. Which statement must be true?

�e quadrilateral panes are squares.

�e quadrilateral panes are rhombuses.

�e triangular panes are all congruent.

�e triangular panes are right triangles.

43. A triangle has side lengths 7 in., 9 in., and x in. Which inequality must be true?

7 x 9 2 x 9 2 x 16 7 x 16

44. ABC and HIG are acute triangles such that ABC HIG. BL and IT are altitudes of the two triangles. Is BL IT ? Justify your answer.

SAT/ACT

hsm11gmse_0906_t14039

A

H

D G C

E B

F

ShortResponse


7 Common Core


content standards 9-4 compositions of g.co.5 . . . specify

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