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Understanding Rotational Transformations in Geometry

This text explores the concept of rotational transformations in geometry, focusing on the motion of an object around a fixed point known as the center of rotation. It discusses key elements such as the direction of rotation (clockwise or counter-clockwise), the angle of rotation, and how these affect the coordinates of the figures. The relationship between figures and their rotated images is analyzed, emphasizing congruence and patterns in the coordinates after transformation. Additionally, it compares clockwise and counter-clockwise rotations to illustrate their equivalence in certain scenarios.

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Understanding Rotational Transformations in Geometry

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  1. Graphing Transformations 3 Rotation – the motion of an object around a fixed point

  2. Direction of rotation – can be clockwise or counter-clockwise

  3. Centre of rotation – the fixed point around which the rotation takes place

  4. Angle of rotation – the amount of rotation made

  5. y 4 3 360° 2 1 x -4 -3 -2 -1 1 2 3 4 -1 -2 centre of rotation -3 -4

  6. y 4 90° counter-clockwise 3 2 +90° 1 x -4 -3 -2 -1 1 2 3 4 -1 -2 270° clockwise -3 -270° -4

  7. y 4 3 180° counter-clockwise +180° 2 1 x -4 -3 -2 -1 1 2 3 4 -1 -2 180° clockwise -3 -180° -4

  8. 4 TOP OF PAGE 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A rotation +90°

  9. 4 TOP OF PAGE 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A rotation +180°

  10. 4 TOP OF PAGE 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A rotation +270°

  11. Would the images be different if the figures had been rotated clockwise instead of counter-clockwise?

  12. 4 TOP OF PAGE 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A rotation -90°

  13. Would the images be different if the figures had been rotated clockwise instead of counter-clockwise? A 90° clockwise rotation produces the same image as a 270° counter-clockwise rotation, and vice versa.

  14. y Coordinates C’ 4 A(0,1) B(2,3) C(4,2) D(3,0) D’ 3 B B’ C 2 A 1 A’ D x -4 -3 -2 -1 1 2 3 4 A’(-1,0) B’(-3,2) C’(-2,4) D’(0,3) -1 -2 -3 -4 A rotation +90° OR-270°

  15. What patterns do you see in the coordinates of the figure and its image? • x-coordinates and y-coordinates switched and the x-coordinates have the opposite sign

  16. y Coordinates 4 A(0,1) B(2,3) C(4,2) D(3,0) 3 B C 2 A 1 D x -4 -3 -2 -1 1 2 3 4 D’ A’(0,-1) B’(-2,-3) C’(-4,-2) D’(-3,0) -1 A’ -2 C’ -3 B’ -4 A rotation +180° OR -180°

  17. What patterns do you see in the coordinates of the figure and its image? • x-coordinates and y-coordinates have the opposite sign

  18. y Coordinates 4 A(0,1) B(2,3) C(4,2) D(3,0) 3 B C 2 A 1 D x -4 -3 -2 -1 1 2 3 4 A’ A’(1,0) B’(3,-2) C’(2,-4) D’(0,-3) -1 -2 B’ -3 D’ C’ -4 A rotation +270° OR -90°

  19. What patterns do you see in the coordinates of the figure and its image? • x-coordinates and y-coordinates switched and the y-coordinates have the opposite sign

  20. How are a figure and its rotation image alike? They are congruent. They have the same orientation. If ABCD is read clockwise, then A’B’C’D’ is read clockwise.

  21. Coordinates y 4 A 3 B D 2 1 x -4 -3 -2 -1 1 2 3 4 C -1 -2 -3 -4 Rotate ABCD -90°

  22. Coordinates y 4 3 2 1 A x B -4 -3 -2 -1 1 2 3 4 D -1 -2 -3 C -4 Rotate ABCD +270°

  23. Coordinates y 4 3 2 A 1 x -4 -3 -2 -1 1 2 3 4 -1 B D -2 C -3 -4 Rotate ABCD 180°

  24. y Coordinates 4 A(0,1) B(2,3) C(4,2) D(3,0) 3 B C 2 A 1 D x -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A rotation

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