# 8.12: Geometry Software and Graphing Rotations

- Page ID
- 6056

Graph rotated images given preimage and number of degrees. Perform rotations using Geogebra.

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.

## Graphs of Rotations

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 \(P′\)) |
---|---|---|---|

\((0, 0)\) | \(90^{\circ}\) (or \(−270^{\circ} ) | \((x,y)\) | \((−y,x)\) |

\((0, 0)\) | \(180^{\circ}\) (or \(−180^{\circ}\) ) | \((x,y)\) | \((−x,−y)\) |

\((0, 0)\) | \(270^{\circ}\) (or \(−90^{\circ}\) ) | \((x,y)\) | \((y,−x)\) |

**Let's draw the preimage and image and properly label each for the following transformation:**

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

**Now, let's draw and label the rotated image for the following rotations: **

- The diamond \(ABCD\) is rotated \(145^{\circ}\) CCW about the origin to form the image \(A′B′C′D′\).

Notice the direction is counter-clockwise.

- The following figure is rotated about the origin \(200^{\circ}\) CW to make a rotated image.

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

Example \(\PageIndex{1}\)

Earlier, you were asked about the 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.

**Solution**

Example \(\PageIndex{2}\)

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.

**Solution**

Notice the direction of the angle is clockwise, therefore the angle measure is \(60^{\circ}\) CW or \(−60^{\circ}\).

Example \(\PageIndex{3}\)

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

**Solution**

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

Example \(\PageIndex{4}\)

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.

**Solution**

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

## Review

- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) clockwise about the origin.
- Rotate the above figure \(270^{\circ}\) clockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

- Rotate the above figure \(90^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(270^{\circ}\) counterclockwise about the origin.
- Rotate the above figure \(180^{\circ}\) about the origin.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 10.8.

## Vocabulary

Term | Definition |
---|---|

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. |

## Additional Resources

Practice: Geometry Software and Graphing Rotations