Lesson 2.4_Rotation of Shapes about an Origin (0,0)

Homework (12/5/16) Transformational Geometry_Rotations Page 1 and 2

https://www.youtube.com/watch?v=VJTxv-tRKj0 http://www.sciencekids.co.nz/gamesactivities/math/transformation.html

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? 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. Opening a drawer? 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. 12/2/2018

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

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

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)

90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2) 12/2/2018

Rotate (-3, -2) 90 clockwise Formula (x, y)  (y, x) A’(-2, 3) (-3, -2) 12/2/2018

90 counter-clockwise rotation Formula (x, y)  (y, x) A’(2, 4) A(4, -2) 12/2/2018

Rotate (-5, 3) 90 counter-clockwise Formula (x, y)  (y, x) (-5, 3) (-3, -5) 12/2/2018

180 rotation Formula (x, y)  (x, y) A’(4, 2) A(-4, -2) 12/2/2018

Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4) 12/2/2018

Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4) Draw ABC A(-3, 0) C(1, -1) 12/2/2018

Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x) 12/2/2018

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) 12/2/2018

Rotate ABC 90 clockwise. Check by rotating ABC 90. A’ B’ A(-3, 0) C’ C(1, -1) 12/2/2018

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 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. –4 What do you notice about each point and its image? B’(–7, –3) –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 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. –4 What do you notice about each point and its image? C’(–4, –4) –5 –6 A’(–7, –6) –7