Geometry Rotations Goals n n Identify rotations in

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Geometry Rotations

Geometry Rotations

Goals n n Identify rotations in the plane. Apply rotation formulas to figures on

Goals n n Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. 11/30/2020

Rotation n A transformation in which a figure is turned about a fixed point,

Rotation n A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation 11/30/2020

Rotation n Rays drawn from the center of rotation to a point and its

Rotation n Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. G 90 Center of Rotation 11/30/2020 G’

A Rotation is an Isometry n n Segment lengths are preserved. Angle measures are

A Rotation is an Isometry n n Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged. 11/30/2020

Rotations on the Coordinate Plane Know the formulas for: • 90 rotations • 180

Rotations on the Coordinate Plane Know the formulas for: • 90 rotations • 180 rotations • clockwise & counterclockwise Unless told otherwise, the center of rotation is the origin (0, 0). 11/30/2020

90 clockwise rotation A(-2, 4) Formula (x, y) (y, x) A’(4, 2) 11/30/2020

90 clockwise rotation A(-2, 4) Formula (x, y) (y, x) A’(4, 2) 11/30/2020

Rotate (-3, -2) 90 clockwise Formula A’(-2, 3) (-3, -2) 11/30/2020 (x, y) (y,

Rotate (-3, -2) 90 clockwise Formula A’(-2, 3) (-3, -2) 11/30/2020 (x, y) (y, x)

90 counter-clockwise rotation Formula A’(2, 4) (x, y) ( y, x) A(4, -2) 11/30/2020

90 counter-clockwise rotation Formula A’(2, 4) (x, y) ( y, x) A(4, -2) 11/30/2020

Rotate (-5, 3) 90 counter-clockwise Formula (-5, 3) (-3, -5) 11/30/2020 (x, y) (

Rotate (-5, 3) 90 counter-clockwise Formula (-5, 3) (-3, -5) 11/30/2020 (x, y) ( y, x)

180 rotation Formula (x, y) ( x, y) A’(4, 2) A(-4, -2) 11/30/2020

180 rotation Formula (x, y) ( x, y) A’(4, 2) A(-4, -2) 11/30/2020

Rotate (3, -4) 180 Formula (-3, 4) (x, y) ( x, y) (3, -4)

Rotate (3, -4) 180 Formula (-3, 4) (x, y) ( x, y) (3, -4) 11/30/2020

Rotation Example B(-2, 4) Draw a coordinate grid and graph: A(-3, 0) C(1, -1)

Rotation Example B(-2, 4) Draw a coordinate grid and graph: A(-3, 0) C(1, -1) B(-2, 4) C(1, -1) Draw ABC 11/30/2020

Rotation Example B(-2, 4) Rotate ABC 90 clockwise. A(-3, 0) 11/30/2020 Formula C(1, -1)

Rotation Example B(-2, 4) Rotate ABC 90 clockwise. A(-3, 0) 11/30/2020 Formula C(1, -1) (x, y) (y, x)

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 11/30/2020

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 11/30/2020 (x, y) (y, x) A(-3, 0) A’(0, 3) B(-2, 4) B’(4, 2) C(1, -1) C’(-1, -1)

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 11/30/2020

Rotate ABC 90 clockwise. B(-2, 4) A’ B’ A(-3, 0) C’ C(1, -1) 11/30/2020 Check by rotating ABC 90.

Rotation Formulas n n 90 CW 90 CCW 180 (x, y) (y, x) (x,

Rotation Formulas n n 90 CW 90 CCW 180 (x, y) (y, x) (x, y) ( y, x) (x, y) ( x, y) Rotating through an angle other than 90 or 180 requires much more complicated math. 11/30/2020

Compound Reflections n If lines k and m intersect at point P, then a

Compound Reflections n If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. 11/30/2020

Compound Reflections n If lines k and m intersect at point P, then a

Compound Reflections n If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. k m P 11/30/2020

Compound Reflections n Furthermore, the amount of the rotation is twice the measure of

Compound Reflections n Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m. k m 45 90 P 11/30/2020

Compound Reflections n The amount of the rotation is twice the measure of the

Compound Reflections n The amount of the rotation is twice the measure of the angle between lines k and m. k m x 2 x P 11/30/2020

Rotational Symmetry n A figure can be mapped onto itself by a rotation of

Rotational Symmetry n A figure can be mapped onto itself by a rotation of 180 or less. 45 90 The square has rotational symmetry of 90. 11/30/2020

Does this figure have rotational symmetry? The hexagon has rotational symmetry of 60. 11/30/2020

Does this figure have rotational symmetry? The hexagon has rotational symmetry of 60. 11/30/2020

Does this figure have rotational symmetry? Yes, of 180. 11/30/2020

Does this figure have rotational symmetry? Yes, of 180. 11/30/2020

Does this figure have rotational symmetry? 90 180 270 360 No, it required a

Does this figure have rotational symmetry? 90 180 270 360 No, it required a full 360 to map onto itself. 11/30/2020

Rotating segments C B A D O H F G 11/30/2020 E

Rotating segments C B A D O H F G 11/30/2020 E

Rotating AC 90 CW about the CE origin maps it to _______. C B

Rotating AC 90 CW about the CE origin maps it to _______. C B A D O H F G 11/30/2020 E

Rotating HG 90 CCW about FE the origin maps it to _______. C B

Rotating HG 90 CCW about FE the origin maps it to _______. C B A D O H F G 11/30/2020 E

Rotating AH 180 about the origin maps it to _______. ED C B A

Rotating AH 180 about the origin maps it to _______. ED C B A D O H F G 11/30/2020 E

Rotating GF 90 CCW about GH point G maps it to _______. C B

Rotating GF 90 CCW about GH point G maps it to _______. C B A D O H F G 11/30/2020 E

Rotating ACEG 180 about the EGAC origin maps it to _______. C C B

Rotating ACEG 180 about the EGAC origin maps it to _______. C C B A A D O H F G G 11/30/2020 E E

Rotating FED 270 CCW about BOD point D maps it to _______. C B

Rotating FED 270 CCW about BOD point D maps it to _______. C B A D O H F G 11/30/2020 E

Summary n n n A rotation is a transformation where the preimage is rotated

Summary n n n A rotation is a transformation where the preimage is rotated about the center of rotation. Rotations are Isometries. A figure has rotational symmetry if it maps onto itself at an angle of rotation of 180 or less. 11/30/2020

Homework 11/30/2020

Homework 11/30/2020