Rotations Geometry ROTATIONS A rotation is an isometric

Rotations Geometry

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.

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

Notation Rcenter, degree

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.

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;

Rotation Definition 1. If point P is NOT point O, then OP = OP’ and m POP’ = Ɵ°.

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.

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

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.

Equivalent Rotations For the rotations below. Give an equivalent rotation. 1. 90° 2. 245° 3. -180° 4. -160° Are there other angles that are equivalent?

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.

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

Co-terminal Angles Find more co-terminal angles of 60°

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.

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

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

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

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. 1. RO, 180° 2. RO, 90° 3. RO, 270°

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. 1. RO, 180° 2. RO, -90° 3. RO, -270°

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)

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