Geometry Rotations

Goals Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. 4/23/2017

Rotation A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation 4/23/2017

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 G’ 4/23/2017

A Rotation is an Isometry Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged. 4/23/2017

Rotations on the Coordinate Plane Know the formulas for: 90 rotations 180 rotations clockwise & counter-clockwise Unless told otherwise, the center of rotation is the origin (0, 0). 4/23/2017

90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2) 4/23/2017

Rotate (-3, -2) 90 clockwise Formula (x, y)  (y, x) A’(-2, 3) (-3, -2) 4/23/2017

90 counter-clockwise rotation Formula (x, y)  (y, x) A’(2, 4) A(4, -2) 4/23/2017

Rotate (-5, 3) 90 counter-clockwise Formula (x, y)  (y, x) (-5, 3) (-3, -5) 4/23/2017

180 rotation Formula (x, y)  (x, y) A’(4, 2) A(-4, -2) 4/23/2017

Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4) 4/23/2017

Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4) Draw ABC A(-3, 0) C(1, -1) 4/23/2017

Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x) 4/23/2017

Rotate ABC 90 clockwise. (x, y)  (y, x) A(-3, 0)  A’(0, 3) B(-2, 4)  B’(4, 2) C(1, -1)  C’(-1, -1) A’ B’ A(-3, 0) C’ C(1, -1) 4/23/2017

Rotate ABC 90 clockwise. Check by rotating ABC 90. A’ B’ A(-3, 0) C’ C(1, -1) 4/23/2017

Rotation Formulas 90 CW (x, y)  (y, x) 90 CCW (x, y)  (y, x) 180 (x, y)  (x, y) Rotating through an angle other than 90 or 180 requires much more complicated math. 4/23/2017

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. 4/23/2017

Compound Reflections k m P 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 4/23/2017

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 4/23/2017

Compound Reflections The amount of the rotation is twice the measure of the angle between lines k and m. k m x 2x P 4/23/2017

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. 4/23/2017

Does this figure have rotational symmetry? The hexagon has rotational symmetry of 60. 4/23/2017

Does this figure have rotational symmetry? Yes, of 180. 4/23/2017

Does this figure have rotational symmetry? 90 180 270 360 No, it required a full 360 to map onto itself. 4/23/2017

Rotating segments A B C D E F G H O 4/23/2017

Rotating AC 90 CW about the origin maps it to _______. CE A B C D E F G H O 4/23/2017

Rotating HG 90 CCW about the origin maps it to _______. FE A B C D E F G H O 4/23/2017

Rotating AH 180 about the origin maps it to _______. ED A B C D E F G H O 4/23/2017

Rotating GF 90 CCW about point G maps it to _______. GH A B C D E F G H O 4/23/2017

Rotating ACEG 180 about the origin maps it to _______. EGAC C G A B C D E F G H A E O 4/23/2017

Rotating FED 270 CCW about point D maps it to _______. BOD A B C D E F G H O 4/23/2017

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. 4/23/2017

Homework 4/23/2017