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]
Do the angles change? If so, how?
Do the segments change? If so, how?
Do the points change? If so, how?
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]
For a 90[math]^\circ[/math] clockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math]
For a 180[math]^\circ[/math] clockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math]
For a 270[math]^\circ[/math] clockwise rotation, the rule for changing each point is [math]\left(x,y\right)\longrightarrow[/math][br][br]
Use the GeoGebra tools to create a polygon and a point at the origin.[br]1.Use this button to create a polygon. [img]data:image/png;base64,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[/img] [br]2. Use this button to plot a point at (0,0).[img]data:image/png;base64,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[/img][br]Then rotate your shape 180˚ counterclockwise.[br]3. Use these buttons to Rotate your polygon. [img]data:image/png;base64,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[/img] then [img]data:image/png;base64,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[/img][br][br]4. Select the image first, then the origin point. It will then prompt you for the angle of rotation. [br]
What counter-clockwise angle of rotation was used to move ABCD to A'B'C'D'?
The challenge today is to explore a variety of compositions of transformations.
Create a triangle with vertices A(1,1) B (2,3) C (4,1).[br]1. Use [img]data:image/png;base64,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[/img] and select polygon to create a triangle with vertices A(1,1) B (2,3) C (4,1).[br][br]Reflect the triangle over the y-axis.[br]2.Use [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAIAAACKIl8oAAACnUlEQVRIib2WQUvjQBTH3ySppbU2HnoQLW3WQymKH6CHaMWLYLwJwoJS/BgFV/biwYN48F49eyyeu6DSowjtJQgmpSBiuzKUZsaYmj1kG1s7WWKt+79l5s1vHi//9xjUaDQopfBp8Tw/Pj4+NjbmriDDMEKh0OfRlmUZhmGa5sTEhLPCjYQLAIIghMPhTqdjmuZf9Ei4Lj0ajbqfo0Q7en5+/iq0K0+0dnR0srWlFYvDs22WCgcHezx/B5ALhWyMmTFMGYbx+PhIKaWUemR9f5/tdCQAIES7vh4uaTY6qyin6fTO1NTS5qaUSg2HRrZtMzdsywKEEMcBQv5xhJB2u+10jeB5p+C55VP/3XzNZjOfz6uq+hk0u9blctmyrFarValUZFnOZDI+cb21ZmddrVYTicT8/Pzq6urT09Nw6bP/VTQa1XW9VqthjGVZvr29BYBU14WUwsbGz3pdX1lZ3N/PBYNsNCNrVVUjkUitVpNlGWMsimI8Hi+VSm7A8fHJ+Xny5ubH4aGGsWfWDHSz2YzH46Io6roOALqui6K4sLBQLpedgFYLACQACUBXVe0D6Gq1KopiMpnEGCuKMjk5eXV1dXl56QasrWUXF3/Nzu7kckvptOSF7nNIoaCdnZ0Gg6Hd3e+C8LtSqdTr9bm5OUVR3h2zbQCwAdC7Vu11SN/km5nJAdwB7K2v7xaLRf8Djzn53hyiaRqlCaeI09MwmOlH9YaWJGl5+dvFxQ7H3W1vl/5xxqf6at2t4EAJfctz8iEEAENCBzXiyWdZ1lehX15egt3GHyWaEEIIcZ99wsPDQyAQGAkXAGKxmLuC2u12b4F65ZjH6QUA4DgOvM0THBiAAs/zPM8zua+vr6ZpEkJM0wwEAs4bFyHk05p/AL8fqCg06gTrAAAAAElFTkSuQmCC[/img] and select "Reflect about line", Click the Triangle, then click the y-axis.[br][br]Rotate the reflected image 90[math]^\circ[/math] counter clockwise.[br](Refer to previous tasks for directions)[br]
What are the new vertices of the reflected then rotated triangle?
A' (-1,-1)[br]B' (-3,-2)[br]C' (-1,-4)