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Rotations California Standards for Geometry 16: Perform basic constructions 17: Prove theorems using coordinate geometry 22: Know the effect of rigid motions on figures in the coordinate plane.

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Properties of a Rotation Rotation –Transformation in which a figure is turned about a fixed point called the CENTER OF ROTATION. –Rays drawn from the center of rotation to a point and its image form the ANGLE OF ROTATION. –Rotations can be clockwise or counterclockwise.

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C If P is not C (the center of rotation), then PC = P’C PP’ xoxo Properties of a Rotation

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If P is C (the center of rotation), then P = P ’ P C P’

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R S P Q T C T’ P’ Q’ R’ S’ Properties of a Rotation

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identify and use rotations C P T Q R S T’ P’ Q’ R’ S’ 88 o

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Rotation Theorem A rotation is an isometry to prove this theorem, you must show that a rotation keeps segment lengths from the preimage to the image this means that AB = A’B’ theorem

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Three Cases are needed to prove that a rotation is an isometry theorem P Q P’ Q’ Case 1: P, Q and C are noncollinear C

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Case 2: P, Q, and C are collinear theorem P Q P’ Q’ C

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Case 3: P and C are the same point theorem P Q P’ Q’ C

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Definition rotation + prop of = Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’ P Q P’ Q’ C

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C.P.C.T.C. Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’ P Q P’ Q’ C

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Example Graph Quad PQRS P (3, 1), Q (4, 0), R (4, 3) S (2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R S (-3, -1) P’

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Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R (-3, -1) P’ Q’ (-4, 0) S

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Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P (-4, -3) Q R Q’ (-3, -1) (-4, 0) P’ R’ S

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(-4, 0) Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R (-4, -3) Q’ (-3, -1) P’ R’ (-2, -4) S’ S

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(-4, 0) Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R (-4, -3) Q’ (-3, -1) P’ R’ S (-2, -4) S’

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theorem Reflection-Rotation Theorem If two lines intersect, then a reflection in the first line followed by a reflection in the second line is the same as a rotation about the point of intersection. B A m P B’ A’ B’’ A’’

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Reflection-Rotation Theorem The angle of rotation is 2 x o, where x o is the measure of the acute or right angle formed by the two lines. theorem xoxo B A P B’ A’ B’’ A’’ 2xo2xo m

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Example is reflected in line k to produce. This triangle is the reflected in line m to produce Describe the transformation k m J K L J’ K’ L’ J” K” L” 45 o P 90 o clockwise rotation

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Rotational Symmetry A figure that can be mapped onto itself by a rotation of 180 o or less. Definition 90 o

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Rotational Symmetry A figure that can be mapped onto itself by a rotation of 180 o or less. Definition 120 o

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Rotational Symmetry A figure that can be mapped onto itself by a rotation of 180 o or less. Definition No rotational symmetry

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Summary What are the properties of a rotation? How are reflections and rotations related? What does it mean when a figure has rotational symmetry?

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