To determine the degree of rotation and direction for the points \( P(-4, 10) \) and \( P'(-10, -4) \), we can use the rotation mapping of coordinates.
The general formulas for rotating a point \( (x, y) \) about the origin by \(\theta\) degrees counterclockwise are:
\[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \]
We are given:
- Initial point \( P(-4, 10) \)
- Rotated point \( P'(-10, -4) \)
To find the angle \(\theta\), we can examine the transformations of the coordinates. Let's find out what happens to \( P \).
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Calculate the original distance of point P from the origin: \[ d_P = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116} \approx 10.77 \]
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Calculate the distance of point P' from the origin: \[ d_{P'} = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \approx 10.77 \]
Both points \( P \) and \( P' \) are the same distance from the origin, confirming that this is a rotation around the origin.
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Next, to find an angle, we will consider the transformation of their coordinates.
Original coordinates: \( P(-4, 10) \)
After rotation: \[ P'(-10, -4) \]
To determine the angle of rotation, look at the changes in their coordinates:
- The x-coordinate changes from \(-4\) to \(-10\)
- The y-coordinate changes from \(10\) to \(-4\)
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Geometric visualization:
- Point \( P \) lies in the second quadrant (negative x, positive y).
- Point \( P' \) lies in the third quadrant (negative x, negative y).
This indicates that the rotation has moved the point from the second quadrant to the third quadrant - typically reflecting a rotation of \( 180^{\circ} \).
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Calculating for the exact angle:
Since the coordinates have shifted from:
- \( x = -4 \) to \( x = -10 \) (which indicates a shift to negative on x)
- \( y = 10 \) to \( y = -4 \) (which indicates a shift to negative on y)
This confirms that the rotation is:
\[ -180^\circ \text{ (clockwise) or } 180^\circ \text{ (counterclockwise)} \]
So the answer is: \[ \text{−180 degrees (clockwise) or 180 degrees (counterclockwise)} \]