35. Think About a Plan The Millenium Wheel, also known as the 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?
36. Reasoning For center of rotation P, does an x ° x degrees  rotation followed by a y ° y degrees  rotation give the same image as a y ° y degrees  rotation followed by an x ° x degrees  rotation? Explain.
37. Writing Describe compositions of rotations that have the same effect as a 360 ° 360 degrees  rotation about a point X.
38. Coordinate Geometry Graph A(5, 2). Graph B, the image of A for a 90 ° 90 degrees  rotation about the origin O. Graph C, the image of A for a 180 ° 180 degrees  rotation about O. Graph D, the image of A for a 270 ° 270 degrees  rotation about O. What type of quadrilateral is ABCD? Explain.

Point O is the center of the regular nonagon below.

39. Find the angle of rotation that maps F to H.
40. Open-Ended Describe a composition of rotations that maps H to C.
41. Error Analysis Your friend says that A B ¯ eh b bar  is the image of E D ¯ e d bar  for a 120 ° 120 degrees  rotation about O. What is wrong with your friend's statement?

In the figure below, 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 ° 30 degrees  angles.)

Image Long Description

42. 120 ° 120 degrees  rotation of B about O
43. 270 ° 270 degrees  rotation of L about O
44. 300 ° 300 degrees  rotation of I B ¯ i b bar  about O
45. 60 ° 60 degrees  rotation of E about O
46. 180 ° 180 degrees  rotation of J K ¯ j k bar  about O
47. 240 ° 240 degrees  rotation of G about O
48. 120 ° 120 degrees  rotation of F about H
49. 270 ° 270 degrees  rotation of M about L
50. 60 ° 60 degrees  rotation followed by 120 ° 120 degrees  rotation of I about O
51. 45 ° 45 degrees  rotation followed by 225 ° 225 degrees  rotation of M about O

C Challenge

52. Reasoning If you are given a figure and a rotation image of the figure, how can you find the center and angle of rotation?
53. Coordinate Geometry Draw Δ L M N cap delta l m n  with vertices L ( 2 , − 1 ) , M ( 6 , − 2 ) , and  N ( 4 , 2 ) . l open 2 comma negative 1 close comma m open 6 comma negative 2 close comma , and n open 4 comma 2 close .  Find the coordinates of the vertices after a 90 ° 90 degrees  rotation about the origin and about each of the points L, M, and N.

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