To determine the rotation of point \( P(-1, 2) \) to point \( P'(2, 1) \), we can analyze the change in coordinates.
- Start with \( P(-1, 2) \).
- After rotating, we end up at \( P'(2, 1) \).
Let's observe the transformation:
- The original coordinates are \( (-1, 2) \).
- The new coordinates are \( (2, 1) \).
To find the rotation, we can consider the origin as the center of rotation for convenience, and calculate the vector from the origin to both points.
- Vector to \( P \): \( (-1, 2) \)
- Vector to \( P' \): \( (2, 1) \)
Next, let's visualize the rotation.
- The point \( P(-1, 2) \) is in the second quadrant and rotates to \( P'(2, 1) \), which is in the first quadrant.
- The vector from the origin to \( P(-1, 2) \) typically points left and up, while the vector to \( P'(2, 1) \) points right and up.
Now let's calculate the angle:
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The angle of the position of \( P(-1, 2) \) can be found using the arctangent function:
- For \( P \): \( \tan^{-1} \left(\frac{2}{-1}\right) \) indicates an angle of \( 180° - \theta \).
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The angle for \( P'(2, 1) \):
- For \( P' \): \( \tan^{-1} \left(\frac{1}{2}\right) \).
Calculating the difference in angles, it can be established through visual examination or through calculation that to rotate \( P(-1, 2) \) to \( P'(2, 1) \), it appears you are rotating 90 degrees counterclockwise (which is a positive rotation).
Thus, the correct answer is:
D. 90° rotation.
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