To determine the degree and direction of rotation for the point \( P(-4, 10) \) to \( P'(-10, -4) \), we can apply the rotation matrix concept or the properties of rotation.
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Find the angle of rotation:
- The coordinates \( P(-4, 10) \) and \( P'(-10, -4) \) represent two positions of the same point.
- To find the angle, recall that a rotation by \( \theta \) degrees transforms a point \( (x, y) \) to \( (x', y') \) using the equations: \[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
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Calculate the angles:
- The original point \( P(-4, 10) \) can be calculated in polar coordinates or assessed geometrically.
- Considering the coordinates, the angle can be derived from their position in the coordinate system.
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Transformation:
- If we analyze the change from \( P \) to \( P' \):
- The x-coordinate transforms from \(-4\) to \(-10\), and the y-coordinate transforms from \(10\) to \(-4\).
- This suggests a rotation that moves the coordinates from the first quadrant (or boundary of the second quadrant) into the third quadrant.
- If we analyze the change from \( P \) to \( P' \):
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Visual Calculation:
- The transformation from the coordinates gives us the angle of rotation.
- A point rotated 90 degrees clockwise from the position in the first quadrant will end up in the fourth quadrant (\(P'\)).
- A further look at the coordinates shows that the movement is essentially \(-90\) degrees (which is a clockwise rotation), matching the point's movement as positive x shifts into negative while y shifts negative from a positive.
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Conclusion:
- The rotation of \( P(-4, 10) \) to \( P'(-10, -4) \) is indeed \( -90 \) degrees clockwise, which is consistent with our calculations.
Thus, the answer is:
−90 degrees (clockwise) or negative 90 degrees (clockwise).