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) 90 degrees (counterclockwise) 90 degrees (counterclockwise) −90 degrees (clockwise) negative 90 degrees (clockwise) −180 degrees (clockwise) or 180 degrees (counterclockwise)

Answer 1

To determine the degree and direction of rotation of the point \( P(-4, 10) \) to the point \( P'(-10, -4) \), we can analyze the transformation using rotation mapping.

Given:
- Original point: \( P(-4, 10) \)
- Rotated point: \( P'(-10, -4) \)

We can use the following rotation formulas about the origin for 90, 180, and 270 degrees:
- **90 degrees counterclockwise**: \( (x, y) \) becomes \( (-y, x) \)
- **180 degrees**: \( (x, y) \) becomes \( (-x, -y) \)
- **270 degrees counterclockwise (or 90 degrees clockwise)**: \( (x, y) \) becomes \( (y, -x) \)

Now, let's apply these transformations to the original point \( P(-4, 10) \):

1. **90 degrees counterclockwise**:
\[
(-4, 10) \rightarrow (-10, -4)
\]
This matches \( P' \).

2. **180 degrees**:
\[
(-4, 10) \rightarrow (4, -10)
\]
This does not match \( P' \).

3. **270 degrees counterclockwise (or 90 degrees clockwise)**:
\[
(-4, 10) \rightarrow (10, 4)
\]
This does not match \( P' \).

Since the 90 degrees counterclockwise rotation gives us \( P'(-10, -4) \), we conclude:

**The point \( P(-4, 10) \) is rotated 90 degrees counterclockwise to become \( P'(-10, -4) \)**.