9.4. Perform Rotations. Draw a rotation. STEP 1 Draw a segment from A to P. STEP 2 Draw a ray to form a 1208 angle with } PA.

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1 erform otations efore You rotated figures about the origin. Now You will rotate figures about a point. Wh? So ou can classif transformations, as in Es Ke Vocabular center of rotation angle of rotation rotation, p. 272 DIETION O OTTION ecall from Lesson 4.8 that a rotation is a transformation in which a figure is turned about a fied point called the center of rotation. as drawn from the center of rotation to a point and its image form the angle of rotation. rotation about a point through an angle of 8 maps ever point Q in the plane to a point Q9 so that one of the following properties is true: If Q is not the center of rotation, then Q 5 Q9 and m QQ9 5 8, or If Q is the center of rotation, then the image of Q is Q. 408 counterclockwise rotation is shown at the right. otations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise ngle of rotation enter of rotation clockwise E X M L E Draw a rotation Draw a 208 rotation of n about. counterclockwise Solution STE Draw a segment from to. STE 2 Draw a ra to form a 208 angle with } STE 3 Draw 9 so that 9 5. STE 4 epeat Steps 3 for each verte. Draw n hapter 9 roperties of Transformations

2 USE OTTIONS You can rotate a figure more than However, the effect is the same as rotating the figure b the angle minus OTTIONS OUT THE OIGIN You can rotate a figure more than 808. The diagram shows rotations of point 308, 2208, and 308 about the origin. rotation of 3608 returns a figure to its original coordinates. There are coordinate rules that can be used to find the coordinates of a point after rotations of 908, 808, or 2708 about the origin KEY ONET or Your Notebook oordinate ules for otations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true:. or a rotation of 908, (a, b) (2b, a). 2. or a rotation of 808, (a, b) (2a, 2b). 3. or a rotation of 2708, (a, b) (b, 2a). (2b, a) (2a, 2b) (a, b) (b, 2a) E X M L E 2 otate a figure using the coordinate rules Graph quadrilateral STU with vertices (3, ), S(5, ), T(5, 23), and U(2, 2). Then rotate the quadrilateral 2708 about the origin. NOTHE WY or an alternative method for solving the problem in Eample 2, turn to page 606 for the roblem Solving Workshop. Solution Graph STU. Use the coordinate rule for a 2708 rotation to find the images of the vertices. (a, b) (b, 2a) (3, ) 9(, 23) S(5, ) S9(, 25) T(5, 23) T9(23, 25) U(2, 2) U9(2, 22) Graph the image 9S9T9U9. at classzone.com S U9 U 9 T T9 S9 GUIDED TIE for Eamples and 2. Trace nde and. Then draw a 508 rotation of nde about. E 2. Graph njkl with vertices J(3, 0), K(4, 3), and L(6, 0). otate the triangle 908 about the origin. D 9.4 erform otations 599

3 USING MTIES You can find certain images of a polgon rotated about the origin using matri multiplication. Write the rotation matri to the left of the polgon matri, then multipl. KEY ONET or Your Notebook otation Matrices (ounterclockwise) 908 rotation 0 2 0G 808 rotation G ED VOULY Notice that a 3608 rotation returns the figure to its original position. Multipling b the matri that represents this rotation gives ou the polgon matri ou started with, which is wh it is also called the identit matri rotation 0 2 0G 3608 rotation 0 0 G E X M L E 3 Use matrices to rotate a figure Trapezoid EGH has vertices E(23, 2), (23, 4), G(, 4), and H(2, 2). ind the image matri for a 808 rotation of EGH about the origin. Graph EGH and its image. VOID EOS ecause matri multiplication is not commutative, ou should alwas write the rotation matri first, then the polgon matri. Solution STE Write the polgon matri: E G H G STE 2 Multipl b the matri for a 808 rotation G E G H G 5 E9 9 G9 H G otation olgon Image matri matri matri STE 3 Graph the preimage EGH. Graph the image E99G9H9. E H9 G9 G H E9 9 GUIDED TIE for Eample 3 Use the quadrilateral EGH in Eample 3. ind the image matri after the rotation about the origin. Graph the image hapter 9 roperties of Transformations

4 THEOEM THEOEM 9.3 otation Theorem rotation is an isometr. roof: Es , p. 604 or Your Notebook n > n 999 SES O THEOEM 9.3 To prove the otation Theorem, ou need to show that a rotation preserves the length of a segment. onsider a segment } Q rotated about point to produce } Q99. There are three cases to prove: ase, Q, and are noncollinear. ase 2, Q, and are collinear. ase 3 and are the same point. E X M L E 4 Standardized Test ractice The quadrilateral is rotated about. What is the value of? 8 } D Solution Theorem 9.3, the rotation is an isometr, so corresponding side lengths are equal. Then 2 5 6, so 5 3. Now set up an equation to solve for (3) Substitute 3 for. 5 2 Solve for. orresponding lengths in an isometr are equal. c The correct answer is.d GUIDED TIE for Eample 4 6. ind the value of r in the rotation of the triangle D 5 2s 4r 2 3 3r s erform otations 60

5 9.4 EXEISES SKILL TIE HOMEWOK KEY 5 WOKED-OUT SOLUTIONS on p. WS for Es. 3, 5, and 29 5 STNDDIZED TEST TIE Es. 2, 20, 2, 23, 24, and 37. VOULY What is a center of rotation? 2. WITING ompare the coordinate rules and the rotation matrices for a rotation of 908. EXMLE on p. 598 for Es. 3 IDENTIYING TNSOMTIONS Identif the tpe of transformation, translation, reflection, or rotation, in the photo. Eplain our reasoning NGLE O OTTION Match the diagram with the angle of rotation at classzone.com OTTING IGUE Trace the polgon and point on paper. Then draw a rotation of the polgon the given number of degrees about G J T S EXMLE 2 on p. 599 for Es. 2 4 USING OODINTE ULES otate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image J K S 2 M L T 602 hapter 9 roperties of Transformations

6 EXMLE 3 on p. 600 for Es. 5 9 USING MTIES ind the image matri that represents the rotation of the polgon about the origin. Then graph the polgon and its image G; J K L G; Q S G; EO NLYSIS The endpoints of } are (2, ) and (2, 3). Describe and correct the error in setting up the matri multiplication for a 2708 rotation about the origin rotation of } 0 2 0G 2 2 3G 2708 rotation of } 2 2 3G 0 2 0G EXMLE 4 on p. 60 for Es MULTILE HOIE What is the value of in the rotation of the triangle about? } 3 D MULTILE HOIE Suppose quadrilateral QST is rotated 808 about the origin. In which quadrant is Q9? I II III D IV INDING TTEN The vertices of n are (2, 0), (3, 4), and (5, 2). Make a table to show the vertices of each image after a 908, 808, 2708, 3608, 4508, 5408, 6308, and 7208 rotation. What would be the coordinates of 9 after a rotation of 8908? Eplain. T S 23. MULTILE HOIE rectangle has vertices at (4, 0), (4, 2), (7, 0), and (7, 2). Which image has a verte at the origin? Translation right 4 units and down 2 units otation of 808 about the origin eflection in the line 5 4 D otation of 808 about the point (2, 0) 24. SHOT ESONSE otate the triangle in Eercise about the origin. Show that corresponding sides of the preimage and image are perpendicular. Eplain. 25. VISUL ESONING point in space has three coordinates (,, z). What is the image of point (3, 2, 0) rotated 808 about the origin in the z-plane? (See Eercise 30, page 585.) HLLENGE otate the line the given number of degrees (a) about the -intercept and (b) about the -intercept. Write the equation of each image ; ; } 2 5; erform otations 603

7 OLEM SOLVING NGLE O OTTION Use the photo to find the angle of rotation that maps onto 9. Eplain our reasoning EVOLVING DOO You enter a revolving door and rotate the door 808. What does this mean in the contet of the situation? Now, suppose ou enter a revolving door and rotate the door What does this mean in the contet of the situation? Eplain. 33. OVING THEOEM 9.3 op and complete the proof of ase. ase The segment is noncollinear with the center of rotation. GIVEN c rotation about maps Q to Q9 and to 9. OVE c Q 5 Q99 STTEMENTS. Q 5 Q9, 5 9, m QQ9 5 m 9 2. m QQ9 5 m Q9 m 9Q9 m 9 5 m Q m Q9 3. m Q9 m 9Q9 5 m Q m Q9 4. m Q 5 m Q99 5.? >? 6. } Q > } Q99 7. Q 5 Q99 ESONS. Definition of? 2.? 3.? ropert of Equalit 4.? ropert of Equalit 5. SS ongruence ostulate 6.? 7.? 9 9 OVING THEOEM 9.3 Write a proof for ase 2 and ase 3. (efer to the diagrams on page 60.) 34. ase 2 The segment is collinear with 35. ase 3 The center of rotation is one the center of rotation. endpoint of the segment. GIVEN c rotation about maps Q to GIVEN c rotation about maps Q Q9 and to 9. to Q9 and to 9., Q, and are collinear. and are the same point. OVE c Q 5 Q99 OVE c Q 5 Q WOKED-OUT SOLUTIONS on p. WS 5 STNDDIZED TEST TIE

8 36. MULTI-STE OLEM Use the graph of a. otate the line 908, 808, 2708, and 3608 about the origin. Describe the relationship between the equation of the preimage and each image. 2 b. Do ou think that the relationships ou described in part (a) are true for an line? Eplain our reasoning. 37. EXTENDED ESONSE Use the graph of the quadratic equation 5 2 at the right. a. otate the parabola b replacing with and with in the original equation, then graph this new equation. 3 b. What is the angle of rotation? c. re the image and the preimage both functions? Eplain. TWO OTTIONS The endpoints of } G are (, 2) and G(3, 4). Graph } 9G9 and } 99G99 after the given rotations. 38. otation: 908 about the origin 39. otation: 2708 about the origin otation: 808 about (0, 4) otation: 908 about (22, 0) 40. HLLENGE polar coordinate sstem locates a point in a plane b its distance from the origin O and b the measure of an angle with its verte at the origin. or eample, the point (2, 308) at the right is 2 units from the origin and m XO What are the polar coordinates of the image of point after a 908 rotation? 808 rotation? 2708 rotation? Eplain X EVIEW repare for Lesson 9.5 in Es MIXED EVIEW In the diagram, ] D is the perpendicular bisector of }. (p. 303) 4. What segment lengths are equal? 42. What is the value of? 43. ind D. (p. 433) D Use a sine or cosine ratio to find the value of each variable. ound decimals to the nearest tenth. (p. 473) 44. v w a b EXT TIE for Lesson 9.4, p. 92 ONLINE QUIZ at classzone.com 605

9 LESSON 9.4 Using LTENTIVE METHODS nother Wa to Solve Eample 2, page 599 MULTILE EESENTTIONS In Eample 2 on page 599, ou saw how to use a coordinate rule to rotate a figure. You can also use tracing paper and move a cop of the figure around the coordinate plane. O L E M Graph quadrilateral STU with vertices (3, ), S(5, ), T(5, 23), and U(2, 2). Then rotate the quadrilateral 2708 about the origin. M E T H O D Using Tracing aper You can use tracing paper to rotate a figure. STE Graph the original figure in the coordinate plane. STE 2 Trace the quadrilateral and the aes on tracing paper. STE 3 otate the tracing paper Then transfer the resulting image onto the graph paper. T I E. GH Graph quadrilateral D with vertices (2, 22), (5, 23), (4, 25), and D(2, 24). Then rotate the quadrilateral 808 about the origin using tracing paper. 2. GH Graph nst with vertices (0, 6), S(, 4), and T(22, 3). Then rotate the triangle 2708 about the origin using tracing paper. 3. SHOT ESONSE Eplain wh rotating a figure 908 clockwise is the same as rotating the figure 2708 counterclockwise. 4. SHOT ESONSE Eplain how ou could use tracing paper to do a reflection. 5. ESONING If ou rotate the point (3, 4) 908 about the origin, what happens to the -coordinate? What happens to the -coordinate? 6. GH Graph njkl with vertices J(4, 8), K(4, 6), and L(2, 6). Then rotate the triangle 908 about the point (2, 4) using tracing paper. 606 hapter 9 roperties of Transformations