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## Rotations

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**Goal**• Identify rotations and rotational symmetry.**Key Vocabulary**• Rotation • Center of rotation • Angle of rotation • Rotational symmetry**Rotation Vocabulary**• Rotation – transformation that turns every point of a pre-image through a specified angle and direction about a fixed point. image Pre-image rotation fixed point**Rotation Vocabulary**• Center of rotation – fixed point of the rotation. Center of Rotation**Rotation Vocabulary**• Angle of rotation – angle between a pre-image point and corresponding image point. image Pre-image Angle of Rotation**Example:**Click the triangle to see rotation Center of Rotation Rotation**Example 1: Identifying Rotations**Tell whether each transformation appears to be a rotation. Explain. B. A. No; the figure appears to be flipped. Yes; the figure appears to be turned around a point.**Your Turn:**Tell whether each transformation appears to be a rotation. b. a. Yes, the figure appears to be turned around a point. No, the figure appears to be a translation.**Rotation Vocabulary**• Rotational symmetry – A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180⁰ or less. Has rotational symmetry because it maps onto itself by a rotation of 90⁰.**Equilateral Triangle**An equilateral triangle has rotational symmetry of order ?**Equilateral Triangle**An equilateral triangle has rotational symmetry of order ?**Equilateral Triangle**3 An equilateral triangle has rotational symmetry of order ? 3 2 1**Hexagon**Regular Hexagon A regular hexagon has rotational symmetry of order ?**Regular Hexagon**A regular hexagon has rotational symmetry of order ?**Regular Hexagon**6 A regular hexagon has rotational symmetry of order ? 5 6 1 4 2 3**Rotational Symmetry**Whena figure can be rotated less than 360° and the image and pre-image are indistinguishable (regular polygons are a great example). Symmetry Rotational: 120° 90° 60° 45°**a.**Rectangle b. Regular hexagon c. Trapezoid SOLUTION a. Yes. A rectangle can be mapped onto itself by a clockwise or counterclockwise rotation of 180° about its center. Example 2 Identify Rotational Symmetry Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself.**b.**Yes. A regular hexagon can be mapped onto itself by a clockwise or counterclockwise rotation of 60°, 120°, or 180° about its center. c. No. A trapezoid does not have rotational symmetry. Example 2 Identify Rotational Symmetry Regular hexagon Trapezoid**Your Turn:**1. Isosceles trapezoid no ANSWER 2. Parallelogram yes; a clockwise or counterclockwise rotation of 180° about its center ANSWER Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself.**Your Turn:**3. Regular octagon yes; a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center ANSWER**Rotation in a Coordinate Plane**• For a Rotation, you need; • An angle or degree of turn • Eg 90° or a Quarter Turn • E.g. 180 ° or a Half Turn • A direction • Clockwise • Anticlockwise • A Centre of Rotation • A point around which Object rotates**8**7 6 y 5 4 3 2 1 x x x x 1 2 3 4 5 6 7 8 –7 –6 –5 –4 –3 –2 –1 -1 x -2 -3 -4 -5 -6 x A Rotation of 90° Counterclockwiseabout (0,0) (x, y)→(-y, x) C(3,5) B’(-2,4) C’(-5,3) B(4,2) A’(-1,2) A(2,1)**8**7 6 y 5 4 3 2 1 x x x x x x x 1 2 3 4 5 6 7 8 –7 –6 –5 –4 –3 –2 –1 -1 x -2 -3 -4 -5 -6 x A Rotation of 180° about (0,0) (x, y)→(-x, -y) C(3,5) B(4,2) A(2,1) A’(-2,-1) B’(-4,-2) C’(-3,-5)**SOLUTION**Plot the points, as shown in blue. Example 4 Rotations in a Coordinate Plane Sketch the quadrilateral with vertices A(2, –2),B(4, 1), C(5, 1), andD(5, –1). Rotate it 90° counterclockwise about the origin and name the coordinates of the new vertices. Use a protractor and a ruler to find the rotated vertices. The coordinates of the vertices of the image are A'(2, 2),B'(–1, 4), C'(–1, 5),andD'(1, 5).**Checkpoint**4. Sketch the triangle with vertices A(0, 0),B(3, 0),andC(3, 4).Rotate∆ABC 90°counterclockwise about the origin. Name the coordinates of the new vertices A',B', and C'. ANSWER Rotations in a Coordinate Plane • A'(0,0), B'(0, 3), C'(–4, 3)