Geometry 4 3 Rotations TopicObjective n n Identify

Geometry 4. 3 Rotations

Topic/Objective n n Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. 12/21/2021 4. 3 Rotations 2

Rotation n A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation 12/21/2021 4. 3 Rotations 3

Rotation n Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. G 90 Center of Rotation 12/21/2021 G’ 4. 3 Rotations 4

A Rotation is an Isometry n n n Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged. Rigid 12/21/2021 4. 3 Rotations 5

Postulate 4. 3 Rotation Postulate n A rotation is a rigid motion 12/21/2021 4. 3 Rotations 6

Rotations on the Coordinate Plane Know the translation formulas for: • 90 rotations • 180 rotations • clockwise & counterclockwise Unless told otherwise, the center of rotation is the origin (0, 0). 12/21/2021 4. 3 Rotations 7

90 clockwise rotation A(-2, 4) Formula (x, y) (y, x) A’(4, 2) 12/21/2021 4. 3 Rotations 8

Rotate (-3, -2) 90 clockwise Formula A’(-2, 3) (x, y) (y, x) (-3, -2) 12/21/2021 4. 3 Rotations 9

90 counter-clockwise rotation Formula A’(2, 4) (x, y) ( y, x) A(4, -2) 12/21/2021 4. 3 Rotations 10

Rotate (-5, 3) 90 counter-clockwise Formula (-5, 3) (x, y) ( y, x) (-3, -5) 12/21/2021 4. 3 Rotations 11

180 rotation Formula (x, y) ( x, y) A’(4, 2) A(-4, -2) 12/21/2021 4. 3 Rotations 12

Rotate (3, -4) 180 Formula (-3, 4) (x, y) ( x, y) (3, -4) 12/21/2021 4. 3 Rotations 13

Rotation Example B(-2, 4) Draw a coordinate grid and graph: A(-3, 0) C(1, -1) B(-2, 4) C(1, -1) Draw ABC 12/21/2021 4. 3 Rotations 14

Rotation Example B(-2, 4) Rotate RST 90 clockwise. A(-3, 0) 12/21/2021 Formula C(1, -1) (x, y) (y, x) 4. 3 Rotations 15

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 12/21/2021 (x, y) (y, x) A(-3, 0) A’(0, 3) B(-2, 4) B’(4, 2) C(1, -1) C’(-1, -1) 4. 3 Rotations 16

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ Check by rotating ABC 90. A(-3, 0) C’ C(1, -1) 12/21/2021 4. 3 Rotations 17

Rotation Formulas n n 90 CW 90 CCW 180 (x, y) (y, x) (x, y) ( y, x) (x, y) ( x, y) Rotating through an angle other than 90 or 180 requires much more complicated math. 12/21/2021 4. 3 Rotations 18

Rotational Symmetry n A figure can be mapped onto itself by a rotation of 180 or less. 45 90 The square has rotational symmetry of 90. 12/21/2021 4. 3 Rotations 19

Does this figure have rotational symmetry? The hexagon has rotational symmetry of 60. 12/21/2021 4. 3 Rotations 20

Does this figure have rotational symmetry? Yes, of 180. 12/21/2021 4. 3 Rotations 21

Does this figure have rotational symmetry? 90 180 270 360 No, it required a full 360 to map onto itself. 12/21/2021 4. 3 Rotations 22

Rotating segments C B A D E O H F G 12/21/2021 4. 3 Rotations 23

Rotating AC 90 CW about the CE origin maps it to _______. C B A D E O H F G 12/21/2021 4. 3 Rotations 24

Rotating HG 90 CCW about FE the origin maps it to _______. C B A D E O H F G 12/21/2021 4. 3 Rotations 25

Rotating AH 180 about the origin maps it to _______. ED C B A D E O H F G 12/21/2021 4. 3 Rotations 26

Rotating GF 90 CCW about GH point G maps it to _______. C B A D E O H F G 12/21/2021 4. 3 Rotations 27

Rotating ACEG 180 about the EGAC origin maps it to _______. C C B A A D E E O H F G G 12/21/2021 4. 3 Rotations 28

Rotating FED 270 CCW about BOD point D maps it to _______. C B A D E O H F G 12/21/2021 4. 3 Rotations 29

Summary n n n A rotation is a transformation where the preimage is rotated about the center of rotation. Rotations are Isometries. A figure has rotational symmetry if it maps onto itself at an angle of rotation of 180 or less. 12/21/2021 4. 3 Rotations 30

Practice 12/21/2021 4. 3 Rotations 31