# Rotations California Standards for Geometry

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Rotations California Standards for Geometry
16: Perform basic constructions 17: Prove theorems using coordinate geometry 22: Know the effect of rigid motions on figures in the coordinate plane.

Properties of a Rotation
Transformation in which a figure is turned about a fixed point called the CENTER OF ROTATION. Rays drawn from the center of rotation to a point and its image form the ANGLE OF ROTATION. Rotations can be clockwise or counterclockwise.

Properties of a Rotation
If P is not C (the center of rotation), then PC = P’C P xo P’ C

Properties of a Rotation
If P is C (the center of rotation), then P = P’ P P’ C

Properties of a Rotation
Q’ Q P’ R S R’ P T’ T S’ C

identify and use rotations
Q’ Q P’ R S R’ P T’ T 88o S’ C

theorem Rotation Theorem A rotation is an isometry
to prove this theorem, you must show that a rotation keeps segment lengths from the preimage to the image this means that AB = A’B’

theorem Three Cases are needed to prove that a rotation is an isometry
Q P P’ Case 1: P, Q and C are noncollinear C Q’

theorem Q Case 2: P, Q, and C are collinear Q’ P’ P C

theorem Case 3: P and C are the same point Q P C P’ Q’

Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’
Definition rotation Definition rotation + prop of =

Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’ C.P.C.T.C. Q P P’

and name the new coordinates
Example S R Graph Quad PQRS P(3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180o counterclockwise about (0, 0) and name the new coordinates P (-3, -1) P’ Q

and name the new coordinates
Example S R Graph Quad PQRS P(3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180o counterclockwise about (0, 0) and name the new coordinates (-4, 0) Q’ P (-3, -1) P’ Q

and name the new coordinates
Example S R Graph Quad PQRS P(3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180o counterclockwise about (0, 0) and name the new coordinates (-4, 0) Q’ P (-3, -1) P’ Q R’ (-4, -3)

and name the new coordinates
Example S R Graph Quad PQRS P(3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180o counterclockwise about (0, 0) and name the new coordinates (-4, 0) Q’ P (-3, -1) P’ Q R’ (-4, -3) S’ (-2, -4)

and name the new coordinates
Example S R Graph Quad PQRS P(3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180o counterclockwise about (0, 0) and name the new coordinates (-4, 0) Q’ P (-3, -1) P’ Q R’ (-4, -3) S’ (-2, -4)

theorem Reflection-Rotation Theorem
If two lines intersect, then a reflection in the first line followed by a reflection in the second line is the same as a rotation about the point of intersection. m A B B’ A’ P B’’ A’’

theorem xo 2xo Reflection-Rotation Theorem
The angle of rotation is 2xo, where xo is the measure of the acute or right angle formed by the two lines. m A 2xo xo B B’ A’ P B’’ A’’

Example is reflected in line k to produce .
This triangle is the reflected in line m to produce Describe the transformation k J’ J” K’ K” K L’ L” 90o clockwise rotation 45o J P L m

Definition 90o Rotational Symmetry
A figure that can be mapped onto itself by a rotation of 180o or less. 90o

120o Definition Rotational Symmetry
A figure that can be mapped onto itself by a rotation of 180o or less. 120o

No rotational symmetry
Definition Rotational Symmetry A figure that can be mapped onto itself by a rotation of 180o or less. No rotational symmetry

Summary What are the properties of a rotation?
How are reflections and rotations related? What does it mean when a figure has rotational symmetry?

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