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# Graphs of Rotations

## Graph rotated images given preimage and number of degrees

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Graphs of Rotations

Quadrilateral WXYZ\begin{align*}WXYZ\end{align*} has coordinates W(5,5),X(2,0),Y(2,3)\begin{align*}W(-5, -5), X(-2, 0), Y(2, 3)\end{align*} and Z(1,3)\begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane. Rotate the image 110\begin{align*}110^\circ\end{align*} counterclockwise about the point X\begin{align*}X\end{align*}. 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\begin{align*}P\end{align*}) Rotated Image (Point P\begin{align*}P^\prime\end{align*})
(0, 0) 90\begin{align*}90^\circ\end{align*}(or 270\begin{align*}-270^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (y,x)\begin{align*}(-y, x)\end{align*}
(0, 0) 180\begin{align*}180^\circ\end{align*}(or 180\begin{align*}-180^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (x,y)\begin{align*}(-x, -y)\end{align*}
(0, 0) 270\begin{align*}270^\circ\end{align*}(or 90\begin{align*}-90^\circ\end{align*}) (x,y)\begin{align*}(x, y)\end{align*} (y,x)\begin{align*}(y, -x)\end{align*}

#### Example A

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

Solution:

#### Example B

The diamond ABCD\begin{align*}ABCD\end{align*} is rotated 145\begin{align*}145^\circ\end{align*}CCW about the origin to form the image ABCD\begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}. 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\begin{align*}200^\circ\end{align*}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\begin{align*}160^\circ\end{align*}.

#### Concept Problem Revisited

Quadrilateral WXYZ\begin{align*}WXYZ\end{align*} has coordinates W(5,5),X(2,0),Y(2,3)\begin{align*}W(-5, -5), X(-2, 0), Y(2, 3)\end{align*} and Z(1,3)\begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane. Rotate the image 110\begin{align*}110^\circ\end{align*} counterclockwise about the point X\begin{align*}X\end{align*}. 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 ST¯¯¯¯¯\begin{align*}\overline{ST}\end{align*} drawn from (-3, 4) to (-3, 8) has been rotated 60\begin{align*}60^\circ\end{align*}CW about the point S\begin{align*}S\end{align*}. Draw the preimage and image and properly label each.

2. The polygon below has been rotated 155\begin{align*}155^\circ\end{align*}CCW about the origin. Draw the rotated image and properly label each.

3. The purple pentagon is rotated about the point A 225\begin{align*}A \ 225^\circ\end{align*}. 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\begin{align*}60^\circ\end{align*}CW or 60\begin{align*}-60^\circ\end{align*}.

2.

Notice the direction of the angle is counter-clockwise, therefore the angle measure is 155\begin{align*}155^\circ\end{align*}CCW or 155\begin{align*}155^\circ\end{align*}.

3.

The measure of BAB=mBAE+mEAB\begin{align*}\angle BAB^\prime = m \angle BAE^\prime + m \angle E^\prime AB^\prime\end{align*}. Therefore BAB=111.80+113.20\begin{align*}\angle BAB^\prime = 111.80^\circ + 113.20^\circ\end{align*} or 225\begin{align*}225^\circ\end{align*}. Notice the direction of the angle is counter-clockwise, therefore the angle measure is 225\begin{align*}225^\circ\end{align*}CCW or 225\begin{align*}225^\circ\end{align*}.

### Practice

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} clockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} clockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} counterclockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} counterclockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} clockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} clockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} counterclockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} counterclockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} clockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} clockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} counterclockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} counterclockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} clockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} clockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} about the origin.

1. Rotate the above figure 90\begin{align*}90^\circ\end{align*} counterclockwise about the origin.
2. Rotate the above figure 270\begin{align*}270^\circ\end{align*} counterclockwise about the origin.
3. Rotate the above figure 180\begin{align*}180^\circ\end{align*} 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.