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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.

### 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.

Answers:

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 \begin{align*}\angle BAB^\prime = 111.80^\circ + 113.20^\circ\end{align*} or \begin{align*}225^\circ\end{align*}. Notice the direction of the angle is counter-clockwise, therefore the angle measure is \begin{align*}225^\circ\end{align*}CCW or \begin{align*}225^\circ\end{align*}.

### Explore More

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

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

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

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

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

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

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

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.8.

### 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.

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