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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.

Then watch this video to see some examples.

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.

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.

Vocabulary Language: English

Rotation

Rotation

A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.