Geometry software has a rotation function that makes it easy to rotate an object about a point. Can you perform a rotation using geometry software without using that function?

### Software for Rotations

Recall that a **rotation** is one example of a **rigid transformation**. A **rotation of \begin{align*}t^\circ\end{align*}** **about a given point \begin{align*}O\end{align*}** takes each point on a shape \begin{align*}P\end{align*} and moves it to \begin{align*}P^\prime\end{align*} such that \begin{align*}P^\prime\end{align*} is on the circle with center \begin{align*}O\end{align*} and radius \begin{align*}\overline{OP}\end{align*} and \begin{align*}\angle P^\prime OP=t^\circ\end{align*}. Below, the parallelogram has been rotated \begin{align*}90^\circ\end{align*} counterclockwise about point \begin{align*}O\end{align*} to create a new parallelogram (the image).

When you are working on a grid (or graph paper), it's possible to perform rotations of \begin{align*}90^\circ, 180^\circ\end{align*}, or \begin{align*}270^\circ\end{align*} by using slopes to find perpendicular lines. But what if the grid is not there? Or the rotation is not a multiple of \begin{align*}90^\circ\end{align*}? Then, it is not as easy to do the rotation because there are no grid lines as a guide.

To perform a rotation without a grid, you need to:

- Construct a circle with center \begin{align*}O\end{align*} through each of the points that define the shape.
- Construct segments connecting each point that defines the shape with point \begin{align*}O\end{align*}.
- Construct an angle of the given number of degrees for the rotation from each segment in the direction specified. The endpoints for these angles should be on the circles previously drawn.
- Connect the endpoints to form the rotated image.

Doing this by hand requires careful construction of circles and angles using a compass and a protractor. Geometry software simplifies this process, because geometry software has a “rotate” button. *Geogebra* is one example of geometry software that is free to download. To perform a rotation in Geogebra, first create your polygon.

Next, create the point you will rotate about (this could be one of the points that defines the shape). You can rename this point if you wish.

Now, rotate the shape about the point by clicking on the “rotate object around point by angle” button, then the shape, then the center of rotation point. Specify the number of degrees and direction in the window that pops up.

Note that the points defining the image are labeled with prime notation. At this point you can move point \begin{align*}O\end{align*} to redefine the center of rotation or change your original shape and your image will move/change accordingly. *Make sure to select the cursor button before attempting to move points.*

Let's look at an example problem.

Show that the center of rotation is the center of circles containing \begin{align*}A\end{align*} and \begin{align*}A^\prime, B\end{align*} and \begin{align*}B^\prime\end{align*}, and \begin{align*}C\end{align*} and \begin{align*}C^\prime\end{align*}.

Construct circles by clicking on the “circle with center through point” button. Then, construct each circle by selecting point \begin{align*}O\end{align*} and then one point on the original triangle.

Notice that the circle containing point \begin{align*}A\end{align*} also contains point \begin{align*}A^\prime\end{align*}, the circle containing point \begin{align*}B\end{align*} also contains point \begin{align*}B^\prime\end{align*}, and the circle containing point \begin{align*}C\end{align*} also contains point \begin{align*}C^\prime\end{align*}.

#### Performing Rotations

Rotate the triangle \begin{align*}90^\circ\end{align*} counterclockwise about point \begin{align*}O\end{align*} without using the rotate button.

First, construct circles with centers at point \begin{align*}O\end{align*} that pass through each of the three points that define the triangle. Use the process from Example A to construct the circles.

Construct segments connecting each point that defines the triangle with the center of the circle.

Next, construct a \begin{align*}90^\circ\end{align*} angle from \begin{align*}\overline{AO}\end{align*} by selecting the “angle with given size” button. Select point \begin{align*}A\end{align*}, then point \begin{align*}O\end{align*}, and then enter \begin{align*}90^\circ\end{align*} counterclockwise in the window that pops up. \begin{align*}A^\prime\end{align*} will appear.

Repeat for each vertex of the triangle. Then, connect \begin{align*}A^\prime\end{align*}, \begin{align*}B^\prime\end{align*} and \begin{align*}C^\prime\end{align*} to form the triangle image.

You could choose to hide the angle markings, circles, and line segments at this point if you wish. *Note that it isn't actually necessary to first construct the circles and line segments in order to do the rotation in this way. However, they allow you to be confident that your rotation is correct because the image points end up on the same circles as their corresponding points. Also, doing it in this way is closest to what you would need to do to perform the construction by hand.*

Compare and contrast the methods for performing rotations in Geogebra explored in the guidance section and Example B.

Both methods correctly performed a rotation. The method in the guidance section was faster. The method in Example B made it more clear that *each* point that defined the triangle was being rotated around a circle with center \begin{align*}O\end{align*}. The method shown in Example B is closer to the method you would have to use to perform a rotation by hand.

**Examples**

**Example 1**

Earlier we talked about using geometry software to perform rotations.

You can perform a rotation using geometry software without the “rotate” button by constructing circles and angles. This method was shown in the second example titled "Performing Rotations".

#### Example 2

Try doing the same rotation without using the rotate button by constructing circles and angles as shown in the second example titled "Performing Rotations". How can you verify that you have done this correctly?

Answers will vary depending on what polygon you construct. Look back at the guidance section for help using Geogebra. To check that you have performed the rotation without the rotate button correctly, just make sure that both images ended up in the same place.

### Review

1. Create a polygon in Geogebra.

2. Rotate the polygon \begin{align*}100^\circ\end{align*} counterclockwise about one of its own points using the rotate button.

3. Rotate the polygon \begin{align*}260^\circ\end{align*} clockwise about the same point that you chose in #2 using the rotate button. What happened? Why?

4. Rotate the polygon \begin{align*}90^\circ\end{align*} counterclockwise about a point that is not on the polygon.

5. Rotate the polygon \begin{align*}90^\circ\end{align*} counterclockwise about the same point you used in #4 without using the rotate button by constructing circles and angles.. Is your rotation correct?

6. Create a regular decagon in Geogebra. *To do this, instead of selecting “polygon”, select “regular polygon”. Plot the first two points of your polygon, then enter the number of points/sides you want your polygon to have (10).*

7. Where would the center of rotation have to be for it to be possible to rotate the decagon less than \begin{align*}360^\circ\end{align*} and have the image be indistinguishable from the original decagon? Find a way to plot this center of rotation.

8. What's the smallest number of degrees you can rotate the decagon about the point from #7 and have the image be indistinguishable from the original decagon? Test this out.

9. Rotations are rigid transformations which means that distance is preserved. Create a new polygon and rotate it about some point. Verify that distance has been preserved by using Geogebra to measure sides of your original polygon and their images. *Select “distance or length” from one of the drop down menus. Then, click on each line segment that you want to measure to see its length.*

10. Rotations are rigid transformations which means that angles are preserved. Verify that angles have been preserved by using Geogebra to measure two corresponding angles for your polygon and image from #9. *Select “angle” from the same drop down menu as in #9. Then, tell Geogebra what angle you want to measure by clicking on the three points you would use to name the angle. You must click on the points in clockwise order for it to measure the correct angle.*

11. Construct a circle in Geogebra.

12. Where would the center of rotation have to be for a rotation of any number of degrees to produce an image that is indistinguishable from the original circle? Test this idea.

13. Could all rotations of circles also have been translations or reflections? Explain.

14. Create a center of rotation that is outside of the circle.

15. Explore how you might be able to rotate the circle \begin{align*}135^\circ\end{align*} counterclockwise about the center of rotation without using the rotate button by constructing additional circles and angles.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.8.