1. Rotations Section 11.8

2. Goal Identify rotations and rotational symmetry.

3. Key Vocabulary Rotation Center of rotation Angle of rotation Rotational symmetry

4. Rotation Vocabulary Rotation – transformation that turns every point of a pre-image through a specified angle and direction about a fixed point. Pre-imageimage fixed point rotation

5. Rotation Vocabulary Center of rotation – fixed point of the rotation. Center of Rotation

6. Rotation Vocabulary Angle of rotation – angle between a pre-image point and corresponding image point. Angle of Rotation image Pre-image

7. Click the triangle to see rotation Center of Rotation Example:

8. Example 1: Identifying Rotations Tell whether each transformation appears to be a rotation. Explain. No; the figure appears to be flipped. Yes; the figure appears to be turned around a point. A. B.

9. Your Turn: Tell whether each transformation appears to be a rotation. No, the figure appears to be a translation. Yes, the figure appears to be turned around a point. a. b.

10. Rotation Vocabulary Rotational symmetry – A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180 ⁰ or less. Has rotational symmetry because it maps onto itself by a rotation of 90 ⁰.

11. Rotational Symmetry When a 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°

12. Identify Rotational Symmetry Example 2 Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself. Regular hexagonb.Rectanglea. Trapezoidc. SOLUTION Yes. A rectangle can be mapped onto itself by a clockwise or counterclockwise rotation of 180° about its center. a.

13. Identify Rotational Symmetry Example 2 Yes. A regular hexagon can be mapped onto itself by a clockwise or counterclockwise rotation of 60°, 120°, or 180° about its center. b. No. A trapezoid does not have rotational symmetry.c. Regular hexagon Trapezoid

14. Your Turn: Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself. Isosceles trapezoid1. Parallelogram2. no ANSWER yes; a clockwise or counterclockwise rotation of 180° about its center ANSWER

15. Your Turn: Regular octagon3. yes; a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center ANSWER

16. Rotating a Figure Rotate quadrilateral VIKE through an angle of 130  abut point Q. V Q V’ Draw segment VQ. Draw circle Q with radius VQ. Measure 130º clockwise with vertex at Q. Mark point D and draw segment QD. I K E Extend segment QD to intersect circle. Mark intersection V’. D You have just rotated V thru a ∠ 130º.

17. Rotating a Figure Rotate quadrilateral VIKE through an angle of 130  abut point Q. V I K E Q V’ E’ Draw segment EQ. Draw circle Q with radius EQ. Measure 130º clockwise with vertex at Q. Mark point D and draw segment QD. Extend segment QD to intersect circle. Mark intersection E’. You have just rotated E thru a ∠ 130º. D

18. Rotating a Figure Rotate quadrilateral VIKE through an angle of 130  abut point Q. V I K E Q V’ K’ E’ Draw segment KQ. Draw circle Q with radius KQ. Measure 130º clockwise with vertex at Q. Mark point D and draw segment QD. Extend segment QD to intersect circle. Mark intersection K’. You have just rotated K thru a ∠ 130º. D

19. Rotating a Figure Rotate quadrilateral VIKE through an angle of 130  abut point Q. V I K E Q V’ I’ K’ E’ Circle Q with radius VQ goes thru I. Draw segment IQ. Measure 130º clockwise with vertex at Q. Mark point D and draw segment QD. Extend segment QD to intersect circle. Mark intersection I’. You have just rotated I thru a ∠ 130º. D

20. Rotating a Figure Rotate quadrilateral VIKE through an angle of 130  abut point Q. V I K E Q V’ I’ K’ E’ Connect point V’ to point I’. Connect point I’ to point K’. Connect point K’ to point E’. Connect point E’ to point V’.

21. Rotations Example 3 Rotate ∆FGH 50° counterclockwise about point C. SOLUTION 1.To find the image of point F, draw and draw a 50° angle. Find F' so that CF = CF'. CF 2.To find the image of point G, draw and draw a 50° angle. Find G' so that CG = CG'. CG

22. Rotations Example 3 3. To find the image of point H, draw and draw a 50° angle. Find H' so that CH = CH'. Draw ∆F'G'H'. CH

23. 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

24. y x 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 -2-2 -3-3 -4-4 -5-5 -6-6 A Rotation of 90° Anticlockwise about (0,0) x x x x x A(2,1) B(4,2) C(3,5) A’(-1,2) B’(-2,4) C’(-5,3) (x, y)→(-y, x)

25. y x 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 -1 -2-2 -3-3 -4-4 -5-5 -6-6 A Rotation of 180° about (0,0) x x x x x x x x A(2,1) B(4,2) C(3,5) (x, y)→(-x, -y) A’(-2,-1) B’(-4,-2) C’(-3,-5)

26. Rotation in a Coordinate Plane

27. Rotations in a Coordinate Plane Example 4 Sketch the quadrilateral with vertices A(2, –2), B(4, 1), C(5, 1), and D(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), and D'(1, 5). SOLUTION Plot the points, as shown in blue.

28. Checkpoint Rotations in a Coordinate Plane Sketch the triangle with vertices A(0, 0), B(3, 0), and C(3, 4). Rotate ∆ABC 90° counterclockwise about the origin. Name the coordinates of the new vertices A', B', and C'. 4. A'(0, 0), B'(0, 3), C'(–4, 3) ANSWER

29. Assignment Pg. 636 – 639; #1 – 41 odd