Review from Friday The composition of two reflections over parallel lines can be described by a translation vector that is: Perpendicular to the two lines Twice the distance between the two lines 1/4/2016

Geometry 9-3 Rotations

1/4/2016 Goals Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane.

1/4/2016 Rotation A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation

1/4/2016 Rotation Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. Center of Rotation 90  G G’

1/4/2016 A Rotation is an Isometry Segment lengths are preserved Angle measures are preserved Parallel lines remain parallel

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).

1/4/  clockwise rotation Formula (x, y)  (y,  x ) A(-2, 4) A’(4, 2)

1/4/2016 Rotate (-3, -2) 90  clockwise Formula (x, y)  (y,  x) (-3, -2) A’(-2, 3)

1/4/  counter-clockwise rotation Formula (x, y)  (  y, x) A(4, -2) A’(2, 4)

1/4/2016 Rotate (-5, 3) 90  counter-clockwise Formula (x, y)  (  y, x) (-3, -5) (-5, 3)

1/4/  rotation Formula (x, y)  (  x,  y) A(-4, -2) A’(4, 2)

1/4/2016 Rotate (3, -4) 180  Formula (x, y)  (  x,  y) (3, -4) (-3, 4)

1/4/2016 Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4) C(1, -1) Draw  ABC A(-3, 0) B(-2, 4) C(1, -1)

1/4/2016 Rotation Example Rotate  ABC 90  clockwise. Formula (x, y)  (y,  x) A(-3, 0) B(-2, 4) C(1, -1)

1/4/2016 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(-3, 0) B(-2, 4) C(1, -1) A’ B’ C’

1/4/2016 Rotate  ABC 90  clockwise. Check by rotating  ABC 90 . A(-3, 0) B(-2, 4) C(1, -1) A’ B’ C’

1/4/2016 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.

1/4/2016 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.

1/4/2016 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. P m k

Compound Reflections Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m. P m k 45  90 

1/4/2016 Compound Reflections The amount of the rotation is twice the measure of the angle between lines k and m. P m k xx 2x 

1/4/2016 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

Homework Page 644 #’s Evens Only