Rotations on a Coordinate Axis
As with translations and reflections, we will now look at rotations on a coordinate axis. For our purposes, we will always use the origin as the center point for our rotations.
To see how this works, let’s look at just two points and the changes in coordinates when rotations are performed.
Start with the point (2, 4) and rotate this point 90° counterclockwise. What is the new point? An easy way to visualize this rotation is simply to rotate your paper.
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90° Counterclockwise Rotation |
As you can see, the new point is (-4, 2). Notice that the new x-coordinate is the opposite of the original y-coordinate and the new y-coordinate is the original x-coordinate.
Now take the same point (2, 4) and rotate it another 90° counterclockwise for a total rotation of 180°. What is the new point?
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180° Counterclockwise Rotation |
As you can see, the new point is (-2, -4). Notice that compared to the original point (2, 4), when a rotation of 180° is applied, the new x-coordinate is the opposite of the original y-coordinate and the new y-coordinate is the opposite of the original x-coordinate. You should also recognize that rotations 180° clockwise and 180° counterclockwise are the same thing.
Now let’s look at rotating the point (2, 4) another 90°. If you have been keeping track, this is a total rotation of 270° counterclockwise. Also, this is the same if you were to rotate 90° clockwise. What is the new point?
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270° Counterclockwise Rotation or 90° Clockwise Rotation |
As you can see, when the point is 270° counterclockwise (90° clockwise) the new point is (4, -2). Notice that the new x-coordinate is the original y-coordinate, and the new y-coordinate is the opposite of the original x-coordinate.
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Summary:
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