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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

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Geometry 9-3 Rotations

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

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

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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’

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

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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).

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1/4/2016 90 clockwise rotation Formula (x, y) (y, x ) A(-2, 4) A’(4, 2)

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

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1/4/2016 90 counter-clockwise rotation Formula (x, y) ( y, x) A(4, -2) A’(2, 4)

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

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1/4/2016 180 rotation Formula (x, y) ( x, y) A(-4, -2) A’(4, 2)

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

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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)

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

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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’

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

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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.

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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.

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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

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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

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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 xx 2x

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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

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Homework Page 644 #’s 14-18 Evens Only

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