To determine the degree of rotation and direction when a point P is rotated about the origin to a new position P', we can apply rotation mapping.
Given:
- Initial point \( P(-4, 10) \)
- Rotated point \( P'(-10, -4) \)
To find the degree and direction of rotation, we can use the rotation formulas:
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A point \( (x, y) \) rotated by an angle \( \theta \) around the origin becomes: \[ (x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)) \]
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We need to find an angle \( \theta \) such that: \[ -10 = -4 \cos(\theta) - 10 \sin(\theta) \] \[ -4 = -4 \sin(\theta) + 10 \cos(\theta) \]
To simplify, let's check existing rotation angles and see which results in the correct coordinates after applying the transformations.
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For 90 degrees counterclockwise (which is \(\theta = 90\)):
- New coordinates: \( (y, -x) \) results in \( (10, 4) \) (not correct).
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For 180 degrees (which is \(\theta = 180\)):
- New coordinates: \( (-x, -y) \) results in \( (4, -10) \) (not correct).
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For 270 degrees counterclockwise (or -90 degrees clockwise):
- New coordinates: \( (-y, x) \) results in \( (-10, -4) \) (correct).
Thus, the point \( P(-4, 10) \) rotates to \( P'(-10, -4) \) through a rotation of 270 degrees counterclockwise or equivalently -90 degrees clockwise.
So the correct answers would be:
- A. 270 degrees (counterclockwise)
- B. -90 degrees (clockwise)
If you need to choose one option from the list provided, you can select either A or B.
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