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Geometry. Rotations. Goals. Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. Rotation. A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation. Rotation.

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## Geometry

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**Geometry**Rotations**Goals**• Identify rotations in the plane. • Apply rotation formulas to figures on the coordinate plane.**Rotation**• A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation**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’**A Rotation is an Isometry**• Segment lengths are preserved. • Angle measures are preserved. • Parallel lines remain parallel. • Orientation is unchanged.**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).**90 clockwise rotation**Formula (x, y) (y, x) A(-2, 4) A’(4, 2)**Rotate (-3, -2) 90 clockwise**Formula (x, y) (y, x) A’(-2, 3) (-3, -2)**90 counter-clockwise rotation**Formula (x, y) (y, x) A’(2, 4) A(4, -2)**Rotate (-5, 3) 90 counter-clockwise**Formula (x, y) (y, x) (-5, 3) (-3, -5)**180 rotation**Formula (x, y) (x, y) A’(4, 2) A(-4, -2)**Rotate (3, -4) 180**Formula (x, y) (x, y) (-3, 4) (3, -4)**Rotation Example**B(-2, 4) Draw a coordinate grid and graph: A(-3, 0) B(-2, 4) C(1, -1) Draw ABC A(-3, 0) C(1, -1)**Rotation Example**B(-2, 4) Rotate ABC 90 clockwise. Formula (x, y) (y, x) A(-3, 0) C(1, -1)**Rotate ABC 90 clockwise.**B(-2, 4) (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)**Rotate ABC 90 clockwise.**B(-2, 4) Check by rotating ABC 90. A’ B’ A(-3, 0) C’ C(1, -1)**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.**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.**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**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**Compound Reflections**• The amount of the rotation is twice the measure of the angle between lines k and m. k m x 2x P**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.**Does this figure have rotational symmetry?**The hexagon has rotational symmetry of 60.**Does this figure have rotational symmetry?**Yes, of 180.**Does this figure have rotational symmetry?**90 180 270 360 No, it required a full 360 to map onto itself.**C**B D A E H F G Rotating segments O**CE**C B D A E H F G Rotating AC 90 CW about the origin maps it to _______. O**FE**C B D A E H F G Rotating HG 90 CCW about the origin maps it to _______. O**ED**C B D A E H F G Rotating AH 180 about the origin maps it to _______. O**GH**C B D A E H F G Rotating GF 90 CCW about point G maps it to _______. O**C**C B D A E A E H F G G Rotating ACEG 180 about the origin maps it to _______. EGAC O**C**B D A E H F G Rotating FED 270 CCW about point D maps it to _______. BOD O**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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