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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.

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**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.

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**Properties of a Rotation**

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

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**Properties of a Rotation**

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

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**Properties of a Rotation**

Q’ Q P’ R S R’ P T’ T S’ C

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**identify and use rotations**

Q’ Q P’ R S R’ P T’ T 88o S’ C

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**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’

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**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’

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

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

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**Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’**

Definition rotation Definition rotation + prop of =

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

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**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

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**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

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**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)

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**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)

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**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)

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**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’’

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**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’’

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**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

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**Definition 90o Rotational Symmetry**

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

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**120o Definition Rotational Symmetry**

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

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**No rotational symmetry**

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

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**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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Review from Friday The composition of two reflections over parallel lines can be described by a translation vector that is: Perpendicular to the two lines.

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