A rotation is a transformation that creates a new figure through "turning" a figure around a given point.[br][br]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."[br]
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 below.
For a 90˚ counterclockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math]
For a 180˚ counterclockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math]
For a 270˚ counterclockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math]
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.
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.
The center of rotation is point R (0,0). Use the GeoGebra tools to find the angle of rotation for these rectangles. Submit your answer below.
What angle of rotation was used to move ABCD to A'B'C'D'?
Use the GeoGebra tools to create a polygon and a point. Then figure out how to rotate your shape 75˚ clockwise.
The challenge today is to explore a variety of compositions of transformations.
Create a polygon and point at the origin to be the center of rotation. [br]Perform a composition of a reflection and rotation. [br]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. [br]Repeat the process with a reflection over the x-axis and a rotation 180˚ counter-clockwise about the origin. [br]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. 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?)
Create a polygon, point at the origin to be the center of rotation, and a vector that will produce the translation [math]\left(x,y\right)\longrightarrow\left(x-2,y-1\right)[/math]. [br]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. [br][br]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. 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?)