To determine the degree of rotation and the direction of rotation from point \(P(-4, 10)\) to \(P'(-10, -4)\), we can use the rotation mapping formulas for points in the Cartesian plane.
The general rotation formulas are:
- For a rotation of \( \theta \) degrees counterclockwise about the origin, the point \((x, y)\) transforms to \((x', y')\) as follows:
\[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
Finding the degree of rotation, let's analyze the original and new coordinates:
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Calculate the distance from the origin for both points:
\[ |P| = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116} \] \[ |P'| = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \]
The distances from the origin remain the same, confirming that it is indeed a rotation.
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Next, determine the angle of rotation by acquiring the angles made by both points with respect to the positive x-axis.
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For \( P(-4, 10) \): \[ \theta_P = \tan^{-1}\left(\frac{10}{-4}\right) \approx \tan^{-1}\left(-2.5\right) \approx 180^\circ - 68.2^\circ \approx 111.8^\circ \]
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For \( P'(-10, -4) \): \[ \theta_{P'} = \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}(0.4) \approx 22.6^\circ \]
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Since \(P'\) is in the third quadrant, we add 180 degrees: \[ \theta_{P'} \approx 180 + 22.6 \approx 202.6^\circ \]
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Now, we can calculate the angle of rotation from \(P\) to \(P'\):
\[ \text{Angle of rotation} = \theta_{P'} - \theta_P \approx 202.6^\circ - 111.8^\circ \approx 90.8^\circ \]
Since the angle is approximately \(90^\circ\) and is positive, this indicates a counterclockwise rotation.
Based on the options given, the closest correct answer is:
90 degrees counterclockwise.