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?

#### Guidance

Recall that a **rotation** is one example of a **rigid transformation**. A **rotation of t∘**

**about a given point**O takes each point on a shape

When you are working on a grid (or graph paper), it's possible to perform rotations of

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

- Construct a circle with center
O through each of the points that define the shape. - Construct segments connecting each point that defines the shape with point
O . - 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 *Make sure to select the cursor button before attempting to move points.*

**Example A**

Show that the center of rotation is the center of circles containing

**Solution:** Construct circles by clicking on the “circle with center through point” button. Then, construct each circle by selecting point

Notice that the circle containing point

**Example B**

Rotate the triangle

**Solution:** First, construct circles with centers at point

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

Next, construct a

Repeat for each vertex of the triangle. Then, connect

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.*

**Example C**

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

**Solution:** 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

**Concept Problem Revisited**

You can perform a rotation using geometry software without the “rotate” button by constructing circles and angles. This method was shown in Example B.

#### Vocabulary

A *rotation of t∘*

**takes each point on a shape**

*about a given point*O

*center of rotation.*
A ** rigid transformation** is a transformation that preserves distance and angles.

#### Guided Practice

1. Create your own polygon and center of rotation in Geogebra.

2. Rotate the polygon

3. Try doing the same rotation without using the rotate button by constructing circles and angles as shown in Example B. How can you verify that you have done this correctly?

**Answers:** Answers will vary depending on what polygon you construct. Look back at the guidance section and Example B 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.

#### Practice

1. Create a polygon in Geogebra.

2. Rotate the polygon

3. Rotate the polygon

4. Rotate the polygon

5. Rotate the polygon

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

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