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

270 degrees (counterclockwise)
270 degrees (counterclockwise)

−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)

help

Answer 1

To determine the degree and direction of rotation from point \( P(-4, 10) \) to point \( P'(-10, -4) \), we can use the rotation mapping method.

1. **Understanding the coordinates**:
- Original point \( P \) is at \( (-4, 10) \).
- Rotated point \( P' \) is at \( (-10, -4) \).

2. **Finding the angle of rotation**:
The general formula for rotating a point \( (x, y) \) by \( θ \) degrees counterclockwise about the origin is given by:
\[
(x', y') = (x \cos θ - y \sin θ, x \sin θ + y \cos θ)
\]
For our specific points, we want to find the \( θ \) such that:
\[
(-10, -4) = (-4 \cos θ - 10 \sin θ, -4 \sin θ + 10 \cos θ)
\]

3. **Using the geometrical interpretation**:
Alternatively, we can visualize the coordinates on a Cartesian plane:
- The point \( P(-4, 10) \) is in the second quadrant.
- The point \( P'(-10, -4) \) is in the third quadrant.

4. **Finding the degree of rotation**:
The rotation from the second quadrant to the third quadrant implies a clockwise rotation. The largest quadrant difference correlates with 270 degrees.
Specifically:
- Starting from \( P \) at negative x and positive y (second quadrant) to a point in the third quadrant (where both x and y are negative) is a clockwise motion.

5. **Conclusion**:
Based on the visual interpretation and quadrant analysis, the point \( P \) has been rotated **270 degrees clockwise**.

Thus, the answer is:

**270 degrees (counterclockwise)**.

Please verify this against the provided options, as the rotation direction might seem unintuitive given the interpretation. If we are strictly using the degree of rotation and denying direction conventions, it is valid to state the absolute magnitude of rotation instead. But sticking with the decision from observation, we conclude:
**270 degrees (counterclockwise)** is still the best fit to describe the situation given back to the set options as they stand.