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
−180
degrees (clockwise) or 180 degrees (counterclockwise)
negative 180 degrees (clockwise) or 180 degrees (counterclockwise)
−90
degrees (clockwise)
negative 90 degrees (clockwise)
90 degrees (counterclockwise)
90 degrees (counterclockwise)
270 degrees (counterclockwise)
and P′(−10,−4)
. Determine the degree of rotation and direction by applying rotation mapping.(1 point)
Responses
−180
degrees (clockwise) or 180 degrees (counterclockwise)
negative 180 degrees (clockwise) or 180 degrees (counterclockwise)
−90
degrees (clockwise)
negative 90 degrees (clockwise)
90 degrees (counterclockwise)
90 degrees (counterclockwise)
270 degrees (counterclockwise)
Answers
GPT-4o mini
To determine the degree of rotation and the direction of rotation of the point from \( P(-4, 10) \) to \( P'(-10, -4) \), we can apply rotation mapping.
1. **Calculate the vector from the origin to the original point and the new point:**
- Vector from origin to \( P \) is \( (-4, 10) \).
- Vector from origin to \( P' \) is \( (-10, -4) \).
2. **Find the angle of rotation:**
- The general formula for rotation by an angle \( \theta \) counterclockwise around the origin is:
\[
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]
3. **Check the transformation of coordinates:**
- The point \( P(-4, 10) \) after a certain rotation results in the point \( P'(-10, -4) \). To find the angle, we can analyze the changes in the coordinates.
4. **Mapping coordinates:**
- We want to find \( \theta \) such that:
\[
(-4, 10) \rightarrow (-10, -4)
\]
5. **Find the angles:**
- The original angle \( \phi \) for point \( P(-4, 10) \) is given by:
\[
\phi = \tan^{-1}\left(\frac{10}{-4}\right) = \tan^{-1}\left(-2.5\right)
\]
- Compute the angle for \( P'(-10, -4) \):
\[
\phi' = \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}\left(0.4\right)
\]
6. **Visualize the movement:**
- From \( P \) to \( P' \), the point has moved from the quadrant II (where \( x \) is negative and \( y \) is positive) to quadrant III (where both \( x \) and \( y \) are negative).
7. **Calculating the rotation:**
- Observing that \( P \) to \( P' \) involves a significant movement through quadrants, we can conclude that the transformation involved a \( 180^\circ \) turn because the signs of both coordinates change.
Hence, the rotation is:
**−180 degrees (clockwise) or 180 degrees (counterclockwise)**.
This confirms that the degree of rotation is 180 degrees, and the direction can be expressed in both clockwise and counterclockwise terminology.
1. **Calculate the vector from the origin to the original point and the new point:**
- Vector from origin to \( P \) is \( (-4, 10) \).
- Vector from origin to \( P' \) is \( (-10, -4) \).
2. **Find the angle of rotation:**
- The general formula for rotation by an angle \( \theta \) counterclockwise around the origin is:
\[
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]
3. **Check the transformation of coordinates:**
- The point \( P(-4, 10) \) after a certain rotation results in the point \( P'(-10, -4) \). To find the angle, we can analyze the changes in the coordinates.
4. **Mapping coordinates:**
- We want to find \( \theta \) such that:
\[
(-4, 10) \rightarrow (-10, -4)
\]
5. **Find the angles:**
- The original angle \( \phi \) for point \( P(-4, 10) \) is given by:
\[
\phi = \tan^{-1}\left(\frac{10}{-4}\right) = \tan^{-1}\left(-2.5\right)
\]
- Compute the angle for \( P'(-10, -4) \):
\[
\phi' = \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}\left(0.4\right)
\]
6. **Visualize the movement:**
- From \( P \) to \( P' \), the point has moved from the quadrant II (where \( x \) is negative and \( y \) is positive) to quadrant III (where both \( x \) and \( y \) are negative).
7. **Calculating the rotation:**
- Observing that \( P \) to \( P' \) involves a significant movement through quadrants, we can conclude that the transformation involved a \( 180^\circ \) turn because the signs of both coordinates change.
Hence, the rotation is:
**−180 degrees (clockwise) or 180 degrees (counterclockwise)**.
This confirms that the degree of rotation is 180 degrees, and the direction can be expressed in both clockwise and counterclockwise terminology.