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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.

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Presentation on theme: "Rotations California Standards for Geometry 16: Perform basic constructions 17: Prove theorems using coordinate geometry 22: Know the effect of rigid motions."— Presentation transcript:

1 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.

2 Properties of a Rotation 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.

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

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

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

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

7 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

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

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

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

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

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

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

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

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

16 (-4, 0) Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R (-4, -3) Q’ (-3, -1) P’ R’ (-2, -4) S’ S

17 (-4, 0) Example Graph Quad PQRS P (3, 1), Q(4, 0), R(4, 3) S(2, 4) and then rotate PQRS 180 o counterclockwise about (0, 0) and name the new coordinates P Q R (-4, -3) Q’ (-3, -1) P’ R’ S (-2, -4) S’

18 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. B A m P B’ A’ B’’ A’’

19 Reflection-Rotation Theorem The angle of rotation is 2 x o, where x o is the measure of the acute or right angle formed by the two lines. theorem xoxo B A P B’ A’ B’’ A’’ 2xo2xo m

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

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

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

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

24 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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