To determine the rotation that transforms the point \( P(-1, 2) \) to \( P'(2, 1) \), we can analyze the positions of the points and the transformations involved.
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Calculate the difference in coordinates: \[ P' - P = (2 - (-1), 1 - 2) = (3, -1) \]
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Based on the rotation options below are the implications of various rotations:
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90 degree rotation: A 90-degree rotation counter-clockwise around the origin transforms a point \( (x, y) \) to \( (-y, x) \).
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-90 degree rotation (or 270 degree rotation clockwise): A -90-degree rotation transforms a point \( (x, y) \) to \( (y, -x) \).
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180 degree rotation: A 180-degree rotation transforms a point \( (x, y) \) to \( (-x, -y) \).
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-270 degree rotation (or 90 degrees rotation clockwise): This is essentially the same as the -90 degrees rotation.
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Identify the transformation: We will examine the original point \( P(-1, 2) \) and apply possible rotations to see which one lands us at \( P'(2, 1) \).
Using the 90-degree rotation: \[ P(-1, 2) \to (-2, -1) \quad \text{(not } P'(2, 1) \text{)} \]
Using the -90 degree rotation: \[ P(-1, 2) \to (2, 1) \quad \text{(this matches } P'(2, 1) \text{)} \]
Using the 180-degree rotation: \[ P(-1, 2) \to (1, -2) \quad \text{(not } P'(2, 1) \text{)} \]
Using the -270 degree rotation: This is equivalent to a 90-degree clockwise rotation, which we have already established corresponds to the same result as the -90 degree rotation.
Given this, the correct rotation that transforms point \( P(-1, 2) \) to \( P'(2, 1) \) is:
-90 degree rotation (or 270 degree clockwise rotation).
1 answer