Rotations. Essential Question How can you rotate a figure in a coordinate plane?

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1 11.3 Rotations Essential Question How can ou rotate a figure in a coordinate plane? Rotating a Triangle in a oordinate lane ONSTRUTING VILE RGUMENTS To be proficient in math, ou need to use previousl established results in constructing arguments Work with a partner. a. Use dnamic geometr software to draw an triangle and label it. b. Rotate the triangle 90 counterclockwise about the origin to form. c. What is the relationship between the coordinates of the vertices of and those of? d. What do ou observe about the side lengths and angle measures of the two triangles? Rotating a Triangle in a oordinate lane Work with a partner. a. The point (, ) is rotated 90 counterclockwise about the origin. Write a rule to determine the coordinates of the image of (, ). b. Use the rule ou wrote in part (a) to rotate 90 counterclockwise about the origin. What are the coordinates of the vertices of the image,? c. raw. re its side lengths the same as those of? Justif our answer. Rotating a Triangle in a oordinate lane Work with a partner. a. The point (, ) is rotated 180 counterclockwise about the origin. Write a rule to determine the coordinates of the image of (, ). Eplain how ou found the rule. b. Use the rule ou wrote in part (a) to rotate (from Eploration ) 180 counterclockwise about the origin. What are the coordinates of the vertices of the image,? ommunicate Your nswer Sample oints (1, 3) (, 3) (, 1) (0, 0) Segments = 3 = = 3.61 ngles m = m = 90 m = How can ou rotate a figure in a coordinate plane? 5. In Eploration 3, rotate 180 counterclockwise about the origin. What are the coordinates of the vertices of the image,? How are these coordinates related to the coordinates of the vertices of the original triangle,? Section 11.3 Rotations 561

2 Lesson What You Will Learn ore Vocabular rotation, p. 56 center of rotation, p. 56 angle of rotation, p. 56 rotational smmetr, p. 565 center of smmetr, p. 565 erform rotations. erform compositions with rotations. Identif rotational smmetr. erforming Rotations ore oncept Rotations rotation is a transformation in which a figure is turned about a fied point called the center of rotation. Ras 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 maps ever point Q in the plane to a point Q so that one of the following R R properties is true. 0 Q If Q is not the center of rotation, then Q = Q and m QQ =, or If Q is the center of rotation, then Q = Q. Q center of rotation angle of rotation irection of rotation clockwise The figure above shows a 0 counterclockwise rotation. Rotations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise unless otherwise noted. rawing a Rotation raw a 10 rotation of about point. counterclockwise SOLUTION Step 1 raw a segment from to. Step raw a ra to form a 10 angle with Step 3 raw so that =. Step Repeat Steps 1 3 for each verte. raw hapter 11 Transformations

3 USING ROTTIONS You can rotate a figure more than 360. The effect, however, is the same as rotating the figure b the angle minus 360. You can rotate a figure more than 180. The diagram shows rotations of point 130, 0, and 310 about the origin. Notice that point and its images all lie on the same circle. rotation of 360 maps a figure onto itself. You can use coordinate rules to find the coordinates of a point after a rotation of 90, 180, or 70 about the origin. ore oncept oordinate Rules for Rotations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true. or a rotation of 90, (a, b) ( b, a). or a rotation of 180, (a, b) ( a, b). ( b, a) (a, b) or a rotation of 70, (a, b) (b, a). ( a, b) 70 (b, a) Rotating a igure in the oordinate lane Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, 3), and U(, 1) and its image after a 70 rotation about the origin. SOLUTION Use the coordinate rule for a 70 rotation to find the coordinates of the vertices of the image. Then graph quadrilateral RSTU and its image. R S (a, b) (b, a) R(3, 1) R (1, 3) S(5, 1) S (1, 5) U U R T 6 T(5, 3) T ( 3, 5) U(, 1) U ( 1, ) T 6 S Monitoring rogress Help in English and Spanish at igideasmath.com 1. Trace E and point. Then draw a 50 rotation of E about point. E. Graph JKL with vertices J(3, 0), K(, 3), and L(6, 0) and its image after a 90 rotation about the origin. Section 11.3 Rotations 563

4 erforming ompositions with Rotations ostulate Rotation ostulate rotation is a rigid motion. E ecause a rotation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the rotation shown. E = E, E = E, = E m = m, m E = m E, m = m ecause a rotation is a rigid motion, the omposition Theorem guarantees that compositions of rotations and other rigid motions, such as translations and reflections, are rigid motions. OMMON ERROR Unless ou are told otherwise, perform the transformations in the order given. erforming a omposition Graph RS with endpoints R(1, 3) and S(, 6) and its image after the composition. Reflection: in the -ais Rotation: 90 about the origin SOLUTION Step 1 Graph RS. Step Reflect RS in the -ais. R S has endpoints R ( 1, 3) and S (, 6). Step 3 Rotate R S 90 about the origin. R S has endpoints R (3, 1) and S (6, ). R ( 1, 3) S (, 6) 6 R(1, 3) R (3, 1) S(, 6) 8 S (6, ) Monitoring rogress Help in English and Spanish at igideasmath.com 3. Graph RS from Eample 3. erform the rotation first, followed b the reflection. oes the order of the transformations matter? Eplain.. WHT I? In Eample 3, RS is reflected in the -ais and rotated 180 about the origin. Graph RS and its image after the composition. 5. Graph with endpoints (, ) and ( 1, 7) and its image after the composition. Translation: (, ) (, 1) Rotation: 90 about the origin 6. Graph TUV with vertices T(1, ), U(3, 5), and V(6, 3) and its image after the composition. Rotation: 180 about the origin Reflection: in the -ais 56 hapter 11 Transformations

5 Identifing Rotational Smmetr figure in the plane has rotational smmetr when the figure can be mapped onto itself b a rotation of 180 or less about the center of the figure. This point is the center of smmetr. Note that the rotation can be either clockwise or counterclockwise. or eample, the figure below has rotational smmetr, because a rotation of either 90 or 180 maps the figure onto itself (although a rotation of 5 does not) The figure above also has point smmetr, which is 180 rotational smmetr. Identifing Rotational Smmetr oes the figure have rotational smmetr? If so, describe an rotations that map the figure onto itself. a. parallelogram b. regular octagon c. trapezoid SOLUTION a. The parallelogram has rotational smmetr. The center is the intersection of the diagonals. 180 rotation about the center maps the parallelogram onto itself. b. The regular octagon has rotational smmetr. The center is the intersection of the diagonals. Rotations of 5, 90, 135, or 180 about the center all map the octagon onto itself. c. The trapezoid does not have rotational smmetr because no rotation of 180 or less maps the trapezoid onto itself. Monitoring rogress Help in English and Spanish at igideasmath.com etermine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself. 7. rhombus 8. octagon 9. right triangle Section 11.3 Rotations 565

6 11.3 Eercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. OMLETE THE SENTENE When a point (a, b) is rotated counterclockwise about the origin, (a, b) (b, a) is the result of a rotation of.. IERENT WORS, SME QUESTION Which is different? ind both answers. What are the coordinates of the vertices of the image after a 90 counterclockwise rotation about the origin? What are the coordinates of the vertices of the image after a 70 clockwise rotation about the origin? What are the coordinates of the vertices of the image after turning the figure 90 to the left about the origin? What are the coordinates of the vertices of the image after a 70 counterclockwise rotation about the origin? Monitoring rogress and Modeling with Mathematics In Eercises 3 6, trace the polgon and point. Then draw a rotation of the polgon about point using the given number of degrees. (See Eample 1.) E E G G R J M K 6 L R 6 Q S T J 566 hapter 11 Transformations In Eercises 7 10, graph the polgon and its image after a rotation of the given number of degrees about the origin. (See Eample.) Q In Eercises 11 1, graph XY with endpoints X( 3, 1) and Y(, 5) and its image after the composition. (See Eample 3.) 11. Translation: (, ) (, + ) Rotation: 90 about the origin 1. Rotation: 180 about the origin Translation: (, ) ( 1, + 1) 13. Rotation: 70 about the origin Reflection: in the -ais 1. Reflection: in the line = Rotation: 180 about the origin

7 In Eercises 15 and 16, graph LMN with vertices L(1, 6), M(, ), and N(3, ) and its image after the composition. (See Eample 3.) 15. Rotation: 90 about the origin Translation: (, ) ( 3, + ) 16. Reflection: in the -ais Rotation: 70 about the origin 6. ( 1, 1) (1, 1) (, 3) (3, ) 7. ONSTRUTION ollow these steps to construct a rotation of b angle around a point O. Use a compass and straightedge. In Eercises 17 0, determine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself. (See Eample.) O REETE RESONING In Eercises 1, select the angles of rotational smmetr for the regular polgon. Select all that appl E G 1 H Step 1 raw,, and O, the center of rotation. Step raw O. Use the construction for coping an angle to cop at O, as shown. Then use distance O and center O to find. Step 3 Repeat Step to find points and. raw. 8. RESONING You enter the revolving door at a hotel. a. You rotate the door 180. What does this mean in the contet of the situation? Eplain. b. You rotate the door 360. What does this mean in the contet of the situation? Eplain. 9. MTHEMTIL ONNETIONS Use the graph of = ERROR NLYSIS In Eercises 5 and 6, the endpoints of are ( 1, 1) and (, 3). escribe and correct the error in finding the coordinates of the vertices of the image after a rotation of 70 about the origin. 5. ( 1, 1) ( 1, 1) (, 3) (, 3) a. Rotate the line 90, 180, 70, and 360 about the origin. Write the equation of the line for each image. escribe the relationship between the equation of the preimage and the equation of each image. b. o ou think that the relationships ou described in part (a) are true for an line that is not vertical or horizontal? Eplain our reasoning. 30. MKING N RGUMENT Your friend claims that rotating a figure b 180 is the same as reflecting a figure in the -ais and then reflecting it in the -ais. Is our friend correct? Eplain our reasoning. Section 11.3 Rotations 567

8 31. RWING ONLUSIONS figure onl has point smmetr. How man rotations that map the figure onto itself can be performed before it is back where it started? 3. NLYZING RELTIONSHIS Is it possible for a figure to have 90 rotational smmetr but not 180 rotational smmetr? Eplain our reasoning. 33. NLYZING RELTIONSHIS Is it possible for a figure to have 180 rotational smmetr but not 90 rotational smmetr? Eplain our reasoning. 3. THOUGHT ROVOKING an rotations of 90, 180, 70, and 360 be written as the composition of two reflections? Justif our answer. 35. USING N EQUTION Inside a kaleidoscope, two mirrors are placed net to each other to form a V. The angle between the mirrors determines the number of lines of smmetr in the mirror image. Use the formula 1 n(m 1) = 180 to find the measure of 1, the angle between the mirrors, for the black glass number n of lines of smmetr. a. b. 38. HOW O YOU SEE IT? You are finishing the puzzle. The remaining two pieces both have rotational smmetr. 1 a. escribe the rotational smmetr of iece 1 and of iece. b. You pick up iece 1. How man different was can it fit in the puzzle? c. efore putting iece 1 into the puzzle, ou connect it to iece. Now how man was can it fit in the puzzle? Eplain. 39. USING STRUTURE 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 (, 30 ) is units from the origin and m XO = 30. What are the polar coordinates of the image of point after a 90 rotation? a 180 rotation? a 70 rotation? Eplain RESONING Use the coordinate rules for counterclockwise rotations about the origin to write coordinate rules for clockwise rotations of 90, 180, or 70 about the origin O 1 X USING STRUTURE XYZ has vertices X(, 5), Y(3, 1), and Z(0, ). Rotate XYZ 90 about the point (, 1) Maintaining Mathematical roficienc The figures are congruent. Name the corresponding angles and the corresponding sides. (Skills Review Handbook) 0. T S Q R Z V Y X W Reviewing what ou learned in previous grades and lessons 1. J M K L 568 hapter 11 Transformations