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Graphs of Rotations

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Graphs of Rotations

Quadrilateral WXYZ has coordinates W(-5, -5), X(-2, 0), Y(2, 3) and Z(-1, 3) . Draw the quadrilateral on the Cartesian plane. Rotate the image 110^\circ counterclockwise about the point X . Show the resulting image.

Watch This

First watch this video to learn about graphs of rotations.

CK-12 Foundation Chapter10GraphsofRotationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10GraphsofRotationsB

Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A rotation is an example of a transformation where a figure is rotated about a specific point (called the center of rotation), a certain number of degrees.

For now, in order to graph a rotation in general you will use geometry software. This will allow you to rotate any figure any number of degrees about any point. There are a few common rotations that are good to know how to do without geometry software, shown in the table below.

Center of Rotation Angle of Rotation Preimage (Point P ) Rotated Image (Point P^\prime )
(0, 0) 90^\circ (or -270^\circ ) (x, y) (-y, x)
(0, 0) 180^\circ (or -180^\circ ) (x, y) (-x, -y)
(0, 0) 270^\circ (or -90^\circ ) (x, y) (y, -x)

Example A

Line \overline{AB} drawn from (-4, 2) to (3, 2) has been rotated about the origin at an angle of 90^\circ CW. Draw the preimage and image and properly label each.

Solution:

Example B

The diamond ABCD is rotated 145^\circ CCW about the origin to form the image A^\prime B^\prime C^\prime D^\prime . On the diagram, draw and label the rotated image.

Solution:

Notice the direction is counter-clockwise.

Example C

The following figure is rotated about the origin 200^\circ CW to make a rotated image. On the diagram, draw and label the image.

Solution:

Notice the direction of the rotation is counter-clockwise, therefore the angle of rotation is 160^\circ .

Concept Problem Revisited

Quadrilateral WXYZ has coordinates W(-5, -5), X(-2, 0), Y(2, 3) and Z(-1, 3) . Draw the quadrilateral on the Cartesian plane. Rotate the image 110^\circ counterclockwise about the point X . Show the resulting image.

Vocabulary

Center of rotation
A center of rotation is the fixed point that a figure rotates about when undergoing a rotation.
Rotation
A rotation is a transformation that rotates (turns) an image a certain amount about a certain point.
Image
In a transformation, the final figure is called the image .
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.

Guided Practice

1. Line \overline{ST} drawn from (-3, 4) to (-3, 8) has been rotated 60^\circ CW about the point S . Draw the preimage and image and properly label each.

2. The polygon below has been rotated 155^\circ CCW about the origin. Draw the rotated image and properly label each.

3. The purple pentagon is rotated about the point A \ 225^\circ . Find the coordinates of the purple pentagon. On the diagram, draw and label the rotated pentagon.

Answers:

1.

Notice the direction of the angle is clockwise, therefore the angle measure is 60^\circ CW or -60^\circ .

2.

Notice the direction of the angle is counter-clockwise, therefore the angle measure is 155^\circ CCW or 155^\circ .

3.

The measure of \angle BAB^\prime = m \angle BAE^\prime + m \angle E^\prime AB^\prime . Therefore \angle BAB^\prime = 111.80^\circ + 113.20^\circ or 225^\circ . Notice the direction of the angle is counter-clockwise, therefore the angle measure is 225^\circ CCW or 225^\circ .

Practice

  1. Rotate the above figure 90^\circ clockwise about the origin.
  2. Rotate the above figure 270^\circ clockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ counterclockwise about the origin.
  2. Rotate the above figure 270^\circ counterclockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ clockwise about the origin.
  2. Rotate the above figure 270^\circ clockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ counterclockwise about the origin.
  2. Rotate the above figure 270^\circ counterclockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ clockwise about the origin.
  2. Rotate the above figure 270^\circ clockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ counterclockwise about the origin.
  2. Rotate the above figure 270^\circ counterclockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ clockwise about the origin.
  2. Rotate the above figure 270^\circ clockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

  1. Rotate the above figure 90^\circ counterclockwise about the origin.
  2. Rotate the above figure 270^\circ counterclockwise about the origin.
  3. Rotate the above figure 180^\circ about the origin.

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