8.12: Geometry Software and Graphing Rotations

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.

Figure $\PageIndex{1}$

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

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

Figure $\PageIndex{2}$

Figure $\PageIndex{3}$

Notice the direction is counter-clockwise.

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

Figure $\PageIndex{4}$

Figure $\PageIndex{5}$

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

Figure $\PageIndex{6}$

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.

Figure $\PageIndex{7}$

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.

Figure $\PageIndex{8}$

Solution

Figure $\PageIndex{9}$

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.

Figure $\PageIndex{10}$

Solution

Figure $\PageIndex{11}$

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

Figure $\PageIndex{12}$

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.

Figure $\PageIndex{13}$

4. Rotate the above figure $90^{\circ}$ counterclockwise about the origin.
5. Rotate the above figure $270^{\circ}$ counterclockwise about the origin.
6. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{14}$

7. Rotate the above figure $90^{\circ}$ clockwise about the origin.
8. Rotate the above figure $270^{\circ}$ clockwise about the origin.
9. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{15}$

10. Rotate the above figure $90^{\circ}$ counterclockwise about the origin.
11. Rotate the above figure $270^{\circ}$ counterclockwise about the origin.
12. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{16}$

13. Rotate the above figure $90^{\circ}$ clockwise about the origin.
14. Rotate the above figure $270^{\circ}$ clockwise about the origin.
15. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{17}$

16. Rotate the above figure $90^{\circ}$ counterclockwise about the origin.
17. Rotate the above figure $270^{\circ}$ counterclockwise about the origin.
18. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{18}$

19. Rotate the above figure $90^{\circ}$ clockwise about the origin.
20. Rotate the above figure $270^{\circ}$ clockwise about the origin.
21. Rotate the above figure $180^{\circ}$ about the origin.

Figure $\PageIndex{19}$

22. Rotate the above figure $90^{\circ}$ counterclockwise about the origin.
23. Rotate the above figure $270^{\circ}$ counterclockwise about the origin.
24. 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