# Rotations

- Author:
- Shannon Rush

- Topic:
- Rotation

A rotation is a transformation that creates a new figure through "turning" a figure around a given point.
The point is called the "center of rotation." Rays drawn from the center of rotation to a point and its image form the "angle of rotation."
Click the Rotation button below to show the rotation of the triangles around the point. Explore the impact of various angles on the image. Then summarize your thinking in your google doc on #1. As you changed the angle of rotation and the vertices of the triangle, what relationships did you observe between the points, segments and angle measures of the original and the imagine created through a rotation?

Now, return to the GeoGebra sketch above. Use the slider to focus on 90˚, 180˚, and 270˚counter-clockwise rotations. Notice how the points change and record your findings in your google doc under #2.

Now, move the center of rotation and observe the effects on the points, segments and angle measures. Adjust the angle of rotation for a variety of centers to have a more complete investigation.
Then summarize your observations in the google doc #3. What did you observe as you moved the center of rotation? Are the effects of the rotation on the points, segments, and angle measures the same or different when the center of rotation is not at the origin? Explain fully.

4. The center of rotation is point R (0,0). Use the GeoGebra tools to find the angle of rotation for these rectangles. Record your answer in the google doc #4.

## Can you do it on your own?

Use the GeoGebra tools to create a polygon and a point. Then figure out how to rotate your shape 75˚ clockwise. Add a screenshot of your completed diagram to #5 in the google doc.

## Challenge 1: Rotations and Reflections

Create a polygon and point at the origin to be the center of rotation.
Perform a composition of a reflection and rotation.
Try reflecting over the y-axis, then rotating the first image 90˚ counter-clockwise about the origin. Then reverse the order, rotating first, then reflecting.
Repeat the process with a reflection over the x-axis and a rotation 180˚ counter-clockwise about the origin.
Continue to explore a variety of compositions of reflections and rotations until you feel like you have tested your observations.
Summarize what you noticed about composition of rotations and reflections in Challenge 1 in the google doc. Are there any interesting combinations? Does the order matter? (In other words, will the same final image be created no matter what order you do the two transformations?)

## Challenge 2: Rotations and Translations

Create a polygon, point at the origin to be the center of rotation, and a vector that will produce the translation .
Perform a composition of a translation and rotation. Start with a rotation of 90˚counter-clockwise about the origin and the above translation. Try both orders and note if the same image is produced.
Continue to explore a variety of compositions of rotations and translations until you feel like you have tested your observations.
Summarize what you noticed about composition of rotations and translations in your google doc. Are there any interesting combinations? Does the order matter? (In other words, will the same final image be created no matter what order you do the two transformations?)

## Challenge!

The challenge today is to explore a variety of compositions of transformations.