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Geometry Rotations Goals Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. 7/6/2017 Rotation A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation 7/6/2017 Rotation 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 7/6/2017 G’ A Rotation is an Isometry Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged. 7/6/2017 Rotations on the Coordinate Plane Know the formulas for: •90 rotations •180 rotations •clockwise & counterclockwise Unless told otherwise, the center of rotation is the origin (0, 0). 7/6/2017 90 clockwise rotation A(-2, 4) Formula (x, y) (y, x) A’(4, 2) 7/6/2017 Rotate (-3, -2) 90 clockwise Formula A’(-2, 3) (-3, -2) 7/6/2017 (x, y) (y, x) 90 counter-clockwise rotation Formula A’(2, 4) (x, y) (y, x) A(4, -2) 7/6/2017 Rotate (-5, 3) 90 counter-clockwise Formula (-5, 3) (-3, -5) 7/6/2017 (x, y) (y, x) 180 rotation Formula (x, y) (x, y) A’(4, 2) A(-4, -2) 7/6/2017 Rotate (3, -4) 180 Formula (-3, 4) (x, y) (x, y) (3, -4) 7/6/2017 Rotation Example B(-2, 4) Draw a coordinate grid and graph: A(-3, 0) A(-3, 0) C(1, -1) B(-2, 4) C(1, -1) Draw ABC 7/6/2017 Rotation Example B(-2, 4) Rotate ABC 90 clockwise. Formula A(-3, 0) 7/6/2017 C(1, -1) (x, y) (y, x) Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 7/6/2017 (x, y) (y, x) A(-3, 0) A’(0, 3) B(-2, 4) B’(4, 2) C(1, -1) C’(-1, -1) Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 7/6/2017 Check by rotating ABC 90. Rotation Formulas 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. 7/6/2017 Compound Reflections If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. 7/6/2017 Compound Reflections If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. k m P 7/6/2017 Compound Reflections Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m. k m 45 90 P 7/6/2017 Compound Reflections The amount of the rotation is twice the measure of the angle between lines k and m. k m x 2x P 7/6/2017 Rotational Symmetry A figure can be mapped onto itself by a rotation of 180 or less. 45 90 The square has rotational symmetry of 90. 7/6/2017 Does this figure have rotational symmetry? The hexagon has rotational symmetry of 60. 7/6/2017 Does this figure have rotational symmetry? Yes, of 180. 7/6/2017 Does this figure have rotational symmetry? 90 180 270 360 No, it required a full 360 to map onto itself. 7/6/2017 Rotating segments C B A D O H F G 7/6/2017 E Rotating AC 90 CW about the origin maps it to _______. CE C B A D O H F G 7/6/2017 E Rotating HG 90 CCW about the origin maps it to _______. FE C B A D O H F G 7/6/2017 E Rotating AH 180 about the origin maps it to _______. ED C B A D O H F G 7/6/2017 E Rotating GF 90 CCW about point G maps it to _______. GH C B A D O H F G 7/6/2017 E Rotating ACEG 180 about the origin maps it to _______. EGAC C C B A A D O H F G G 7/6/2017 E E Rotating FED 270 CCW about BOD point D maps it to _______. C B A D O H F G 7/6/2017 E Summary 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. 7/6/2017 Homework 7/6/2017