To determine the angle of rotation that takes point \( P(-4, 10) \) to point \( P'(-10, -4) \), we can analyze their positions in relation to the origin.
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Identify the original and new coordinates:
- \( P(x_1, y_1) = (-4, 10) \)
- \( P'(x_2, y_2) = (-10, -4) \)
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Determine the direction and angle of rotation: To find the angle of rotation, we will use the formula for rotation of a point around the origin. The general rotation transformation is given by:
\[ \begin{pmatrix} x' \ y' \end{pmatrix}
\begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
We will check the coordinates:
After rotation by \( \theta \):
- For point \( P(-4, 10) \):
\[ x' = -4 \cos \theta + 10 \sin \theta \] \[ y' = -4 \sin \theta + 10 \cos \theta \]
This should equal \( P'(-10, -4) \): \[ -4 \cos \theta + 10 \sin \theta = -10 \quad (1) \] \[ -4 \sin \theta + 10 \cos \theta = -4 \quad (2) \]
- Substituting values and calculating the angle \(\theta\):
Let's analyze the coordinates geometrically:
- The original point \( P(-4, 10) \) is in the second quadrant.
- The new point \( P'(-10, -4) \) is in the third quadrant.
Finding the angle of rotation:
We can calculate the angle from the origin to both points.
For point P:
\[ \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{10}{-4}\right) = \tan^{-1}\left(-\frac{5}{2}\right) \approx 180 - \theta \quad (\text{since it's in the second quadrant}) \]
So, the angle for \( P \) is approximately \( 180 - \theta \).
For point P':
\[ \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}\left(\frac{2}{5}\right) \text{ (since both coordinates are negative, it's in the third quadrant)}. \]
Thus, the angle corresponding to \( P' \) is \( 180 + \tan^{-1}\left(\frac{2}{5}\right)\).
Calculating rotation angle:
The angle moved from \( P \) to \( P' \) can be computed.
We find:
- From initial approximate angle values, \( P \approx 180^\circ - \theta \)
- \( P' \approx 180^\circ + \tan^{-1}\left(\frac{2}{5}\right) + 180^\circ \)
Verifying angle shift:
Calculating the shift gives you \( 180^\circ \).
Final Result:
The rotation from \( P \) to \( P' \) can be confirmed as \( 180^\circ \) counterclockwise or \( -180^\circ \) clockwise.
Conclusion:
Thus, the answer is:
180 degrees (counterclockwise) or −180 degrees (clockwise).