1. Lesson 2.4_Rotation of Shapes about an Origin (0,0)

2. Homework (12/5/16)
Transformational Geometry_Rotations Page 1 and 2

5. Can you suggest any other examples?
Rotation
Which of the following are examples of rotation in real life?
Opening a door?
Walking up stairs?
Riding on a Ferris wheel?
Bending your arm?
Opening your mouth?
Anything that is fixed at a point and turns about that point is an example of a rotation. This is true even if a complete rotation cannot be completed, such as your jaw when opening your mouth.
Opening a drawer?
Can you suggest any other examples?

6. 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

7. A Rotation is an Isometry
Segment lengths are preserved.
Angle measures are preserved.
Parallel lines remain parallel.
Orientation is unchanged.
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8. Rotation
In order to rotate an object we need 3 pieces of information
Center of rotation
Angle of rotation (degrees)
Direction of rotation

9. Rotation Vocabulary
Center of rotation – fixed point of the rotation. It can be any point on the coordinate plane
Center of Rotation

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

11. Rotation Vocabulary
Angle of rotation – angle between a pre-image point and corresponding image point. It will be in degrees (basic degrees that we will focus: 90 degrees, 180 degrees, and 270 degrees).
image
Pre-image
Angle of Rotation

12. Rotation Vocabulary
Direction of rotation– it will be counter-clockwise or clockwise

13. 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.

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

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

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

17. 90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2)
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18. Rotate (-3, -2) 90 clockwise
Formula
(x, y)  (y, x)
A’(-2, 3)
(-3, -2)
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19. 90 counter-clockwise rotation
Formula
(x, y)  (y, x)
A’(2, 4)
A(4, -2)
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20. Rotate (-5, 3) 90 counter-clockwise
Formula
(x, y)  (y, x)
(-5, 3)
(-3, -5)
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21. 180 rotation
Formula
(x, y)  (x, y)
A’(4, 2)
A(-4, -2)
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22. Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4)
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23. Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4)
Draw ABC
A(-3, 0)
C(1, -1)
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24. Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x)
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25. Rotate ABC 90 clockwise.
(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)
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26. Rotate ABC 90 clockwise.
Check by rotating ABC 90. A’ B’
A(-3, 0) C’
C(1, -1)
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27. Rotation in a Coordinate Plane

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. Rotations on a coordinate grid
The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). 7
A(2, 6) 6 5
B(7, 3) 4 3
C’(–4, 1) 2
Rotate the triangle 180° clockwise about the origin and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2
C(4, –1) –3
Pupils should notice that when a shape is rotated through 180º about the origin, the x-coordinate of each image point is the same as the x-coordinate of the the original point × –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. In other words the coordinates are the same, but the signs are different. –4
What do you notice about each point and its image?
B’(–7, –3) –5 –6
A’(–2, –6) –7

30. Rotations on a coordinate grid
The vertices of a triangle lie on the points A(–6, 7), B(2, 4) and C(–4, 4). 7
B(2, 4) 6 5
C(–4, 4) 4 3
B’(–4, 2) 2
Rotate the triangle 90° anticlockwise about the origin and label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 –3
Pupils should notice that when a shape is rotated through 90º anticlockwise about the origin, the x-coordinate of each image point is the same as the y-coordinate of the the original point × –1. The y-coordinate of the image point is the same as the x-coordinate of the original point. –4
What do you notice about each point and its image?
C’(–4, –4) –5 –6
A’(–7, –6) –7