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Lesson 9-3 Rotations 561

9-3

Objective To draw and identify rotation images of figures

Rotations

In the diagram, the point (3, 2) is rotated counterclockwise about the origin. The point (x1, y1) is the result of a 90˚ rotation. The point (x2, y2) is the result of a 180˚ rotation, and the point (x3, y3) is the result of a 270˚ rotation. What are the coordinates of (x1, y1), (x2, y2), and (x3, y3)? What do you notice about how the coordinates of the points relate to the coordinates (3, 2) after each rotation?

geom12_se_ccs_c09l03_t06.ai

(3, 2)

(x3, y3 )

(x1, y1 )

(x2, y2 )

x

y

Notice the position of the point, in relation to the x- and y-axis, as it rotates around the origin.

In the Solve It, you thought about how the coordinates of a point change as it turns, or rotates, about the origin on a coordinate grid. In this lesson, you will learn how to recognize and construct rotations of geometric figures.

Essential Understanding Rotations preserve distance, angle measures, and orientation of figures.

Lesson Vocabulary

•rotation•center of rotation•angle of rotation

LessonVocabulary Key Concept Rotation About a Point

A rotation of x° about a point Q, called the center of rotation, is a transformation with these two properties:• TheimageofQ is itself (that is, Q′ = Q).• ForanyotherpointV, QV ′ = QV and

m∠VQV ′ = x.

Thenumberofdegreesafigure rotates is the angle of rotation.

A rotation about a point is a rigid motion. You write the x° rotation of △UVW about point Q as r(x°, Q)(△UVW) = △U′V′W′.

V�

Q�

Q

W�

U�

U

WV

x �

The preimage V andits image V�areequidistant fromthe center of rotation.

hsm11gmse_0903_t08068.aiUnless stated otherwise, rotations in this book are counterclockwise.

G-CO.A.4 Develop definitions of rotations . . . in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Also G-CO.A.2, G-CO.B.6

MP 1, MP 3, MP 4

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 1

562 Chapter 9 Transformations

Drawing a Rotation Image

What is the image of r(100°, C)(△LOB)?

Step 1 Draw CO. Use a protractor to draw a 100° angle with vertex C and side CO.

Step 2 Use a compass to construct CO′ ≅ CO.

Step 3 Locate B9 and L9 in a similar manner.

Step 4 Draw △L′O′B′.

1. Copy △LOB from Problem 1. What is the image of △LOB for a 50° rotation about B?

When a figure is rotated 90°, 180°, or 270° about the origin O in a coordinate plane, you can use the following rules.

L

O

C

B

hsm11gmse_0903_t08070.ai

L

O

C

100�

B


L

O

C

O�

B


L

O

C

B�

L�O�

B

hsm11gmse_0903_t08073.aihsm11gmse_0903_t08074.ai

L

O

C

B�

L�O�

B

Got It?

Key Concept Rotation in the Coordinate Plane

r(90°, O)(x, y) = (-y, x) r(180°, O)(x, y) = (-x, -y)

r(270°, O)(x, y) = (y, -x) r(360°, O)(x, y) = (x, y)


2 4 6�2

�2

�4�6x

2

4y

O

G (2, 3)G� (�3, 2)


G (2, 3)

G� (�2, �3)

180� 2 4 6�2�4�6

x

2

�2

4y


2 4 6

�2

�2�4�6x

2

4y

270�

G (2, 3)

G� (3, �2)


42 6�2

�2

�4�6x

2

4y

G� (2, 3)

360�

How do you use the definition of rotation about a point to help you get started?You know that O and O9 must be equidistant from C and that m∠OCO′ must be 100.

Problem 2

Problem 3


Drawing Rotations in a Coordinate Plane

PQRS has vertices P(1, 1), Q(3, 3), R(4, 1), and S(3, 0). What is the graph of r(90°, O)(PQRS).

First,graphtheimagesofeachvertex.

P′ = r(90°, O)(1, 1) = (-1, 1)

Q′ = r(90°, O)(3, 3) = (-3, 3)

R′ = r(90°, O)(4, 1) = (-1, 4)

S′ = r(90°, O)(3, 0) = (0, 3)

Next, connect the vertices to graph P′Q′R′S′.

2. Graph r(270°, O)(FGHI).

You can use the properties of rotations to solve problems.

Using Properties of Rotations

In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of the opposite sides of the parallelogram are equal?

Because T is the midpoint of the diagonals, XT = ZT and WT = YT. Since W and Y are equidistant from T, and the measure of ∠WTY = 180, you know that r(180°, T)(W) = Y . Similarly, r(180°, T)(X) = Z .

You can rotate every point on WX in this same way, so r(180°, T)(WX) = YZ .

Likewise, you can map WZ to YX with r(180°, T)(WZ) = YX .

Because rotations are rigid motions and preserve distance, WX = YZ and WZ = YX .

3. Can you use the properties of rotations to prove that WXYZ is a rhombus? Explain.


2 4 6

�2

�4 �2�6x

4y

R

SO

P

Q

P� (�1, 1)

Q� (�3, 3)

R� (�1, 4)S� (0, 3)


42�4�6

�4

2

4y

F (�3, 2)

I (0, 1)

H (�1, �1)G (�3, �1)

x6

Got It?


W Z

YX

T

Got It?

How do you know where to draw the vertices on the coordinate plane?Use the rules for rotating a point and apply them to each vertex of the figure. Then graph the points and connect them to draw the image.

What do you know about rotations that can help you show that opposite sides of the parallelogram are equal?You know that rotations are rigid motions, so if you show that the opposite sides can be mapped to each other, then the side lengths must be equal.

Lesson Check


Practice and Problem-Solving Exercises

Copy each figure and point P. Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image.

9. 60° 10. 90° 11. 180° 12. 90°

Copy each figure and point P. Then draw the image of JK for a 180° rotation about P. Use prime notation to label the vertices of the image.

13. 14. 15. 16.

PracticeA See Problem 1.


B

A

D

P


R

P

E

T C


DP

R

B


T

KP


J P K


JP

K


J

K

P


K

J � P

Do you know HOW? 1. Copy the figure and point P. Draw r(70°, P)(△ABC).

A

B P

C


In the figure below, point A is the center of square SQRE.

2. What is r(90°, A)(E)?

3. What is the image of RQ for a 180° rotation about A?

4. Use the properties of rotations to describe how you know that the lengths of the diagonals of the square are equal.

Do you UNDERSTAND? 5. Vocabulary △A′B′C′ is a rotation image of

△ABC about point O. Describe how to find the angle of rotation.

6. Error Analysis A classmate drew a 115° rotation of △PQR about point P, as shown at the right. Explain and correct your classmate’s error.

7. Compare and Contrast Compare rotating a figure about a point to reflecting the figure across a line. How are the transformations alike? How are they different?

8. Reasoning Point P(x, y) is rotated about the origin by 135° and then by 45°. What are the coordinates of the image of point P? Explain


S Q

E R

A

P P

R

R

Q

Q

115





For Exercises 17–19, use the graph at the right.

17. Graph r(90°, O)(FGHJ).



20. Thecoordinatesof△PRS are P(-3, 2), R(2, 5), and S(0, 0). What are the coordinates of the vertices of r(270°, O)(△PRS)?

21. V′W′X′Y′ has vertices V′(-3, 2), W′(5, 1), X′(0, 4), and Y′(-2, 0). If r(90°, O)(VWXY) = V′W′X′Y′, what are the coordinates of VWXY?

22. Ferris Wheel AFerriswheelisdrawnonacoordinateplanesothatthefirstcarislocated at the point (30, 0). What are the coordinates of the first car after a rotation of 270°about the origin?

For Exercises 23–25, use the diagram at the right. TQNV is a rectangle. M is the midpoint of the diagonals.

23. Use the properties of rotations to show that the measures of both pairs of opposite sides are equal in length.

24. Reasoning Can you use the properties of rotations to show that the measures of the lengths of the diagonals are equal?

25. Reasoning Can you use properties of rotations to conclude that the diagonals of TQNV bisect the angles of TQNV? Explain.

26. In the diagram at the right, M′N′ is the rotation image of MN about point E. Name all pairs of angles and all pairs of segments that have equal measures in the diagram.

27. Language Arts Symbols are used in dictionaries to help users pronounce wordscorrectly.Thesymbol is called a schwa. It is used in dictionaries to represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What transformation maps a to a lowercase e?

Find the angle of rotation about C that maps the black figure to the blue figure.

28. 29. 30.

See Problem 2.


4�4�6

�4

2

4y

O

F (0, 3) J (3, 2)

H (1, �4)

G (�4, 1)x

6

See Problem 3.


NM

T

V

Q

ApplyB


M

N

E

N’M’


C


C


C


Car 3

Car 18

Car 3

Car 18

31. Think About a Plan TheMilleniumWheel,alsoknownasthe London Eye, contains 32 observation cars. Determine the angle of rotation that will bring Car 3 to the position of Car 18.

• How do you find the angle of rotation that a car travels when it moves one position counterclockwise?

• How many positions does Car 3 move?

32. Reasoning ForcenterofrotationP, does an x° rotation followed by a y° rotation give the same image as a y° rotation followed by an x° rotation? Explain.

33. Writing Describe how a series of rotations can have the same effect as a 360° rotation about a point X.

34. Coordinate Geometry Graph A(5, 2). Graph B, the image of A for a 90° rotation about the origin O. Graph C, the image of A for a 180° rotation about O. Graph D, the image of A for a 270° rotation about O. What type of quadrilateral is ABCD? Explain.

Point O is the center of the regular nonagon shown at the right.

35. FindtheangleofrotationthatmapsF to H.

36. Open-Ended Describe a rotation that maps H to C.

37. Error Analysis Your friend says that AB is the image of ED for a 120° rotation about O. What is wrong with your friend’s statement?

In the figure at the right, the large triangle, the quadrilateral, and the hexagon are regular. Find the image of each point or segment for the given rotation or composition of rotations. (Hint: Adjacent green segments form 30° angles.)

38. r(120°, O)(B) 39. r(270°, O)(L)

40. r(300°, O)(IB) 41. r(60°, O)(E)

42. r(180°, O)(JK) 43. r(240°, O)(G)

44. r(120°, H)(F) 45. r(270°, L)(M)

46. r(180°, O)(I) 47. r(270°, O)(M)

48. Coordinate Geometry Draw △LMN with vertices L(2, -1), M(6, -2), and N(4, 2). Findthecoordinatesoftheverticesaftera90° rotation about the origin and about each of the points L, M, and N.

49. Reasoning If you are given a figure and a rotation image of the figure, how can you find the center and angle of rotation?

A B

C

DO

EF

G

H

I



K

J

I

A

B

DG

F E

L

MH CO

ChallengeC


Apply What You’ve Learned

Look back at the information about the video game on page 543. 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 in Lesson 9-2, you looked at how some translations and reflections move the puzzle piece. Now you will look at how rotations move the puzzle piece.

a. How does the orientation of the puzzle piece compare to the orientation of the target area?

b. Can you move the puzzle piece to the target area using only rotations? Explain.

Forparts(c)–(e),copythegraphandthengraphtheimageof△ABC for the given rotation.

c. r(90°, O)(x, y) d. r(180°, O)(x, y) e. r(270°, O)(x, y)

PERFO

RMANCE TA

SK MATHEMATICAL PRACTICESMP 5

Standardized Test Prep

50. What is the image of (1, -6) for a 90° counterclockwise rotation about the origin?

(6, 1) (-1, 6) (-6, -1) (-1, -6)

51. Thecostumecrewforyourschoolmusicalmakesaprons like the one shown. If blue ribbon costs $1.50 per foot, what is the cost of ribbon for six aprons?

$15.75 $42.00

$31.50 $63.00

52. In △ABC, m∠A + m∠B = 84. Which statement must be true?

BC 7 AC AC 7 BC AB 7 BC BC 7 AB

53. Use the following statement: If two lines are parallel, then the lines do not intersect. a. What are the converse, inverse, and contrapositive of the statement? b. What is the truth value of each statement you wrote in part (a)? If a statement is

false, give a counterexample.

SAT/ACT

hsm11gmse_0903_t14047

18 in.

24 in.

5 in.

5 in.

5 in.

5 in.

ShortResponse

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