93 Rotations

Mathematics Florida StandardsMAFS.912.G-C0.1.4 Develop definitions of rotations...in terms of angles, circles, perpendicular lines, parallellines, and line segments. Also MAFS.912.G-C0.1.2,MAFS.912.G-C0.2.6

MP1,MP 3, MP4

Objective To draw and identify rotation images of figures

f _Getting Ready! X C 4

Notice the

position of thepoint, in relation tothe X- and y-axis,as it rotates

ground the origin.

In the diagram, the point (3, 2) Is rotatedcounterclockwise about the origin. The point(xj, yi) Is the result of a 90** rotation. The point(xg, yz) is the result of a 180" rotation, and thepoint (X3, Vs) Is the result of a 270" rotation.What are the coordinates of (xj. yj), (xg, yz),and (X3, y3)? What do you notice about how thecoordinates of the points relate to the coordinates(3, 2) after each rotation?

zl,

MATHEMATICAL

PRACTICES

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 torecognize and construct rotations of geometric figures.

Essential Understanding Rotations preserve distance, angle measures, andorientation of figures.

Lesson ^Vocabularyrotation

• center of rotation

• angle of rotation

Q'

Key Concept Rotation About a Point

A rotation of .v° about a point Q, called the

center of rotation, is a transformation with

these two properties:

• The image of Q is itself (that is, Q' = Q).• For any other point V, QV = QV and

m^VQV'=x.

The number of degrees a figure

rotates is the angle of rotation.

A rotation about a point is a rigid motion.

You write the x° rotation of AUVW about

point Q as q)(AUVW] = hU'V"W'.

\ The preimage Vandits image V'areequidistant from

I the center of rotation.

Unless stated otherwise, rotations in this book are counterclockwise.

c PowerGeometry.com Lesson 9-3 Rotations 561

Problem 1 Drawing a Rotation Image

What is the image of r^oo",

How do you use thedefinition of rotation

about a point to helpyou get started?You know that 0 and 0'

must be equidistant fromC and that mLOCO'

must be 100.

Step 1

Draw CO. Use a

Step 2

Use a compass

Step 3 Step 4

protractor to draw a to constructLocate B' and L' in a Draw AL'O'B'.

similar manner.

100° angle with

vertex C and side CO.

CO' = CO.

Vs./ N

0(C oV.,

B

Got It? 1. Copy ALOB from Problem 1. What is the image of ALOB for a 50° rotationabout B1

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

''(90°, o)(^' y) = i~y'

Key Concept Rotation in the Coordinate Plane

''(180°, Op, y) = {-X, -y)y

(2 3Ij'(H3. 2.)

■)< f

Y X

-£ ' 0 2 1

y

(2, 3)r

/

/

1 80'/ t X

-£ -4 > V 2 i f5

{6 2 3)'

'■(270°, = ''(36o°,o)U'y) = (^'y)A y 1

,5 (2, 3)

—

/ t ' ■ X

-£ — i -1 2 i £m\ X

X

G' (3

A y

0 )

X/

X

-f — i ' 2 ,■

3

o

562 Chapter 9 Transformations

How do you knowwhere to draw the

vertices on the

coordinate plane?Use the rules for rotatinga point and apply themto each vertex of the

figure. Then graph thepoints and connect themto draw the image.

Problem 2 Drawing Rotations in a Coordinate Plane

PQRS has vertices P(l, 1), Q(3,3), R(4,1), and S(3,0).What is the graph of r^go®, q^{PQRS).

First, graph the images of each vertex.

P' ='-(90=.o)a'l) = (-l'l)

Q' ~ '"(90°, 0)(^' ~ C~3, 3)

~ '"(90°, O)^^' ~

~ ̂90°, 0)(^' ~

Next, connect the vertices to graph P'Q'i?'S'.

Got It? 2. Graph r(27o=, o)(^GH/). A y

F-3,r

./ n

' 7u

X

-6 1 i / ) 1

-G-[-3, rr-: )- Ui-1

What do you knowabout rotations that

can help you showthat opposite sidesof the parallelogramare equal?You know that rotations

are rigid motions, soif you show that theopposite sides can bemapped to each other,then the side lengthsmust be equal.

You can use the properties of rotations to solve problems.

Using Properties of RotationsProblem 3

In the diagram, WXYZ is a parallelogram, and Tis the midpoint ofthe diagonals. How can you use the properties of rotations to showthat 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 Wand Yare equidistant from T, and the measure of Z. WTY = 180,

you know that r^go'. d(^ = Similarly, r){X) = Z.

You can rotate every point on WX in this same way, so r^gg®^ 7^(WX) = YZ.Likewise, you can map WZ to YX with rj-jgg®^ 7)(1YZ) = YX.

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

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

C PowerGeometry.com Lesson 9-3 Rotations 563

Lesson Check

Do you know HOW?

1. Copy the figure andpointP. Draw

In the figure below, point A is the center of

square SQRE.

2. What isr(9o=^)(£)?

3. What is the image of RQ for a

180° rotation about

4. Use the properties of rotations

to describe how you know that

the lengths of the diagonals of

the square are equal.

MATHEMATICAL

Do you UNDERSTAND? PRACTICES

5. Vocabulary AA 'B'C is a rotation image ofAABC about point O. Describe how to find the

angle of rotation.

6. Error Analysis A classmatedrewa 115° rotation of APQR

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(A:,y) is rotated about the originby 135° and then by 45°. What are the coordinates

of the image of point P? Explain

MATHEMATICAL

PRACTICESPractice and Problem-Solving Exercises

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

9.60° 10.90° 11.180°

D

T

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.

^ See Problem 1.

12. 90

K

13.K

15. W 16. t/C

U = P

564 Chapter 9 Transformations

Apply

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

17. Graph r^QQ^ Q^iFGHf).

18. Graph r^igo". o)(^GH/).

19. Graph

^ See Problem 2.

-1 ! i6(-4,i) ̂ 2

1-6 -4 0

-4

y

j(3,2)

/ X

/1 f

ii/ i

4)

20. The coordinates of APRS are

P{-S, 2), Ri2,5), and S(0,0). Whatare the coordinates of the vertices of

'"(270°,

21. V'W'X'Y' hasvertices V'{-3,2), 1), r(0,4), and r(-2,0).If r(9o° ojCVWXY) = V WX'y, what are the coordinates of VWXYl

22. Ferris Wheel A Ferris wheel is drawn on a coordinate plane so that the first car is

located at the point (30, 0). What are the coordinates of the first car after a rotationof 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 ofopposite sides are equal in length.

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

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

^ See Problem 3.

26. In the diagram at the right, M'N' is the rotation image of MN aboutpoint E. Name all pairs of angles and all pairs of segments that have equalmeasures in the diagram.

27. Language Arts Symbols are used in dictionaries to help users pronounce

words correctly. The symbol a is called a schwa. It is used in dictionaries torepresent neutral vowel sounds such as a in ago, i in sanity, and u infocus.What transformation maps a a to a lowercase e?

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

M' N

28.

c

n.

c

U

II

c PowerGeometry.com I Lesson 9-3 Rotations 565

31. Think About a Plan The Millenium Wheel, also known as theLondon Eye, contains 32 observation cars. Determine the angleof 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 For center of rotation P, does an yf rotation

followed by a y° rotation give the same image as a y° rotationfollowed by an xf rotation? Explain.

33. Writing Describe how a series ofrotations can have the same

effect as a 360° rotation about a point X

34. Coordinate Geometry Graph/l(5, 2). Graph B, the image ofA for a 90° rotation about the origin O. Graph C, the image ofA for a 180° rotation about O. Graph D, the image of A for a270° rotation about O. What type of quadrilateral is ABCD"?Explain.

m

Y2

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

35. Find the angle of rotation that maps f to H.

36. Open-Ended Describe a rotation that maps HtoC.

37. Error Analysis Your friend says that AB is the image of ED for a 120° rotationabout 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 orsegment for the given rotation or composition of rotations.

{Hint: Adjacent green segments form 30° angles.)

38. o)(^) 39- '(270°, o)(^5

'"(300°, 0)CfS) too", 0)C£}

^2- ̂ (180°, o)t/^ ^3- ''(240°,

'■(120°,/flW ^5. '■(270°,46. /"(igoo.OlW 47. r(27o=, o)M

^Achallenge 48. Coordinate Geometry Draw ALMAf with vertices Z,(2,-1), M(6,-2), andN(4,2).Find the coordinates of the vertices after a 90° rotation about the origin and abouteach of the points L, M, and N.

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

566 Chapters Transformations

^^/Aa

Sfandardized Test Prep

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

Ca:^ (6, 1) CT^ (-1,6) CO (-6, -1) CO (-1. -6)

Short

. ResponseNi-*

51. The costume crew for your school musical makes

aprons like the one shown. If blue ribbon costs $1.50

per foot, what is the cost of ribbon for she aprons?

CD S15.75 CD $42.00

CO $31.50 CD $63.00

52. In AABC, mAA + mLB = 84. Which statement must be true?

ca:)Bc>ac co^c>bc coab>bc

18 in.

5 in

5,n.A

24 in.

CO-BOAB

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 isfalse, give a counterexample.

Apply What You've Learned MATHEMATICAL

PRACTICES

MR 5

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.

4

y

A\ ; iS'

11 i^ 1C| X

—ii > 0-F-

-2-t

D

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.

For parts (c)-(e), copy the graph and then graph the image of AABC for the given rotation.

c- ̂ (90°, op' y) d. r(l8o^ op> y) e. r(27o=, o)U t)

C PowerGeometry.com Lesson 9-3 Rotations 567