In previous activities, you played with isometries that were created for you. To answer questions in this activity, you will have to use GeoGebra tools to create your own isometries. Use your constructions to support your answers.[br][br][i]If you need to type greek letters, pres Alt (or Ctrl on Macs) + keyboard letters. For example, Alt + b will insert [/i][math]\beta[/math][i] (on a PC).[/i]
In other words, is composition of translation and rotation commutative?[br][br]1a. In the applet below, provide a clear visual argument to support your answer. Use a slider to define a rotation.[br][br]1b. Is there a special case (certain specific translations or rotations), for which the composition would be commutative?
It is not. What would the vector have to be for it to be commutative?
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In other words, is composition of translation and reflection commutative?[br][br]2a. In the applet below, provide a clear visual argument to support your answer. [br][br]2b. Is there a special case (certain specific translations or reflections), for which the composition would be commutative?[br][br]2c. Use your answer in 2b to discuss the relationship between a "Slide flip" and Glide Reflection.
It is not. It is possible to make it commutative by placing the vector at a special position with respect to the line. What is the position?
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In other words, is composition of reflection and rotation commutative?[br][br]3a. In the applet below, provide a clear visual argument to support your answer. Use a slider to define a rotation.[br][br]3b. Is there a special case (certain specific reflections or rotations), for which the composition would be commutative?
It is not, unless the center of rotation is on the line of reflection and the rotation angle 180°.
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4a. In the applet below, provide a clear visual argument to support your answer.[br][br]4b. Is there a special case, for which the composition would be commutative?
It is not, unless the lines are at a special position. What is the position?
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5a. In the applet below, provide a clear visual argument to support your answer. If needed, use a slider to define a rotation.
Yes, there are. Think of composing two isometries of the same kind.