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.
image
Pre-image
rotation
fixed point

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

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

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

8. 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.

9. 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.

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 180⁰ or less.
Has rotational symmetry because it maps onto itself by a rotation of 90⁰.

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

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

14. Your Turn:
Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself.
Isosceles trapezoid 1. no
ANSWER
Parallelogram 2.
yes; a clockwise or counterclockwise rotation of 180° about its center
ANSWER

15. Your Turn: Regular octagon 3.
yes; a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center
ANSWER 15

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

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

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

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

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

21. Rotate ∆FGH 50° counterclockwise about point C.
Example 3
Rotations
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. Example 3
Rotations 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 (x, y)→(-y, x) A Rotation of 90° Anticlockwise about (0,0) 8 7 6 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

25. 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)

26. Rotation in a Coordinate Plane

27. 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), and D(5, –1). Rotate it 90° counterclockwise about the origin and name the coordinates of the new vertices.
SOLUTION
Plot the points, as shown in blue.
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).

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