1. Rotations
Advanced Geometry

2. ROTATIONS
A rotation is an isometric transformation in which a figure is turned about a fixed point. The fixed point is the center of rotation.

3. Angle of Rotation
Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.

4. Notation
Rcenter, degree

5. Full Rotation
One full rotation is ______°, this would return all points in the plane to their original location. Because a rotation can go in two directions along the same arc we need to define positive and negative rotation values.

6. Rotation Definition
A rotation about a Point O through Ɵ degrees is an isometric transformation that maps every point P in the plane to a point P’, so that the following properties are true;

7. Rotation Definition If point P is NOT point O, then
OP = OP’ and mPOP’ = Ɵ°.

8. Rotation Definition
2. If point P IS point O, then P = P’. The center of rotation is the ONLY point in the plane that is unaffected by a rotation.

9. Rotation Direction
Counterclockwise rotation is a positive direction! Clockwise rotation is a negative direction!

10. Equivalent Rotations
Because angles are formed along an arc of a circle there are two ways to get to the same location, a positive direction and a negative direction.

11. Equivalent Rotations
For the rotations below. Give an equivalent rotation ° ° ° ° Are there other angles that are equivalent?

12. Co-terminal Angles
Co-terminal angles are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side.

13. Co-terminal Angles
Co-terminal angles can be calculated using the formula, Co-terminal Angle = Initial Angle + 360n Where n is an integer.

14. Find more co-terminal angles of 60°

15. Applying Rotations
We can use some tricks to rotate figures 90°, 180°, and 270° about the origin. Use the tricks to find the new coordinates and plot.

16. Rotation Tricks
Rotation of 90° or -____ ° about the origin can be described as (x, y)  (-y, x)

17. Rotation Tricks
Rotation of 270° or -____ ° about the origin can be described as (x, y)  (y, -x)

18. Rotation Tricks
Rotation of 180° about the origin can be described as (x, y)  (-x, -y)

19. Example
A quadrilateral has vertices A(-2, 0), B(-3, 2), C(-2, 4), and D(-1, 2). Give the vertices of the image after the described rotations.
RO, 180°
RO, 90°
RO, 270°

20. Example
A triangle has vertices F(-3, 3), G(1, 3), and H(1, 1). Give the vertices of the image after the described rotations.
RO, 180°
RO, -90°
RO, -270°

21. Isometric Properties
Because a rotation is an isometric transformation the following properties are preserved between the pre-image and its image:
Distance (lengths of segments are the same)
Angle measure (angles stay the same)
Parallelism (things that were parallel are still parallel)
Collinearity (points on a line, remain on the line)

22. Transformation Properties
Because a rotation is a transformation that maps all points along an arc the following properties are present.
Distances are different
Orientation is the same
Special Point – Center of Rotation

23. Transformation Properties
Because a rotation is a transformation that maps all points along an arc the following properties are present.

24. Transformation Properties
Because a rotation is a transformation that maps all points along an arc the following properties are present.