Question
A point is rotated about the origin. Its coordinates are P(−4,10) and P′(−10,−4) . Determine the degree of rotation and direction by applying rotation mapping.(1 point) Responses −90 degrees (clockwise) negative 90 degrees (clockwise) −180 degrees (clockwise) or 180 degrees (counterclockwise) negative 180 degrees (clockwise) or 180 degrees (counterclockwise) 90 degrees (counterclockwise) 90 degrees (counterclockwise) 270 degrees (counterclockwise)
Answers
GPT-4o mini
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.
1. **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:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
2. **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.
3. **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.
4. **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.
5. **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).
1. **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:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
2. **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.
3. **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.
4. **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.
5. **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).