Homework Monday 5/23: Rotation of Shapes page 1

Rotation of Shapes about an Origin (0,0)

Can you suggest any other examples? Rotation Which of the following are examples of rotation in real life? Opening a door? Walking up stairs? Riding on a Ferris wheel? Bending your arm? Opening your mouth? Opening a drawer? Anything that is fixed at a point and turns about that point is an example of a rotation. This is true even if a complete rotation cannot be completed, such as your jaw when opening your mouth. Can you suggest any other examples?

Rotation Vocabulary Rotation – transformation that turns every point of a pre-image through a specified angle and direction about a fixed point. image Pre-image rotation fixed point

A Rotation is an Isometry Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged. 9/21/2018

Rotation In order to rotate an object we need 3 pieces of information Center of rotation Angle of rotation (degrees) Direction of rotation

Rotation Vocabulary Center of rotation – fixed point of the rotation. It can be any point on the coordinate plane Center of Rotation

Rotation G 90 Center of Rotation G’ 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 G’ 9/21/2018

Rotation Vocabulary Angle of rotation – angle between a pre-image point and corresponding image point. It will be in degrees (basic degrees that we will focus: 90 degrees, 180 degrees, and 270 degrees) image Pre-image Angle of Rotation

Rotation Vocabulary Direction of rotation– it will be counter-clockwise or clockwise

Rotation Example: Click the triangle to see rotation Center of Rotation Rotation

Example 1: Identifying Rotations Tell whether each transformation appears to be a rotation. Explain. B. A. No; the figure appears to be flipped. Yes; the figure appears to be turned around a point.

Your Turn: Tell whether each transformation appears to be a rotation. Yes, the figure appears to be turned around a point. No, the figure appears to be a translation.

y x A Rotation of 90° Anticlockwise about (0,0) 8 7 6 C(3,5) 5 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 –7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6 A Rotation of 90° Anticlockwise about (0,0) (x, y)→(-y, x) x x x C(3,5) x B’(-2,4) C’(-5,3) B(4,2) A’(-1,2) A(2,1) x

y x 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 –7 –6 –5 –4 –3 –2 –1 -1 -2 -3 -4 -5 -6 A Rotation of 180° about (0,0) (x, y)→(-x, -y) x x x x C(3,5) x B(4,2) x A(2,1) x x A’(-2,-1) B’(-4,-2) C’(-3,-5)

Rotations on the Coordinate Plane Know the formulas for: 90 rotations 180 rotations clockwise & counter-clockwise Unless told otherwise, the center of rotation is the origin (0, 0). 9/21/2018

90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2) 9/21/2018

Rotate (-3, -2) 90 clockwise Formula (x, y)  (y, x) A’(-2, 3) (-3, -2) 9/21/2018

90 counter-clockwise rotation Formula (x, y)  (y, x) A’(2, 4) A(4, -2) 9/21/2018

Rotate (-5, 3) 90 counter-clockwise Formula (x, y)  (y, x) (-5, 3) (-3, -5) 9/21/2018

180 rotation Formula (x, y)  (x, y) A’(4, 2) A(-4, -2) 9/21/2018

Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4) 9/21/2018

Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4) Draw ABC A(-3, 0) C(1, -1) 9/21/2018

Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x) 9/21/2018

Rotate ABC 90 clockwise. (x, y)  (y, x) A(-3, 0)  A’(0, 3) B(-2, 4)  B’(4, 2) C(1, -1)  C’(-1, -1) A’ B’ A(-3, 0) C’ C(1, -1) 9/21/2018

Rotate ABC 90 clockwise. Check by rotating ABC 90. A’ B’ A(-3, 0) C’ C(1, -1) 9/21/2018

Rotation in a Coordinate Plane

Checkpoint Rotations in a Coordinate Plane Sketch the triangle with vertices A(0, 0), B(3, 0), and C(3, 4). Rotate ∆ABC 90° counterclockwise about the origin. Name the coordinates of the new vertices A', B', and C'. 4. A'(0, 0), B'(0, 3), C'(–4, 3) ANSWER

Rotations on a coordinate grid The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). 7 A(2, 6) 6 5 B(7, 3) 4 3 C’(–4, 1) 2 Rotate the triangle 180° clockwise about the origin and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 C(4, –1) –3 –4 What do you notice about each point and its image? B’(–7, –3) Pupils should notice that when a shape is rotated through 180º about the origin, the x-coordinate of each image point is the same as the x-coordinate of the the original point × –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. In other words the coordinates are the same, but the signs are different. –5 –6 A’(–2, –6) –7

Rotations on a coordinate grid The vertices of a triangle lie on the points A(–6, 7), B(2, 4) and C(–4, 4). 7 B(2, 4) 6 5 C(–4, 4) 4 3 B’(–4, 2) 2 Rotate the triangle 90° anticlockwise about the origin and label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 –3 –4 What do you notice about each point and its image? Pupils should notice that when a shape is rotated through 90º anticlockwise about the origin, the x-coordinate of each image point is the same as the y-coordinate of the the original point × –1. The y-coordinate of the image point is the same as the x-coordinate of the original point. C’(–4, –4) –5 –6 A’(–7, –6) –7