To determine the degree and direction of rotation of point P(−4, 10) to point P′(−10, −4) about the origin, we can apply the concept of rotation mapping.
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Finding the angle of rotation: We start with the original point P(−4, 10) and see how it transforms into point P′(−10, −4).
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Understanding the rotation transformation: A rotation in a 2D plane about the origin can be represented using rotation matrices or geometric reasoning.
- A 90 degrees counterclockwise rotation transforms coordinates (x, y) to (-y, x).
- A 90 degrees clockwise rotation transforms (x, y) to (y, -x).
- A 180 degrees rotation transforms (x, y) to (-x, -y).
- A 270 degrees counterclockwise rotation or a 90 degrees clockwise rotation switches the transformations.
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Translating the coordinates: Let's check how point P(−4, 10) would be positioned if we rotated it through the various degrees:
- 90 degrees counterclockwise: \[ (−4, 10) \rightarrow (−10, −4) \] This matches the point P′.
This means the transformation from P to P′ is indeed a 90-degree counterclockwise rotation.
Thus, the degree of rotation is 90 degrees counterclockwise.
Therefore, the correct choice is:
90 degrees (counterclockwise).