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)

Answer 1

To determine the degree and direction of rotation of point P(−4, 10) to point P′(−10, −4) about the origin, we can apply the concept of rotation mapping.

1. **Finding the angle of rotation**:
We start with the original point P(−4, 10) and see how it transforms into point P′(−10, −4).

2. **Understanding the rotation transformation**:
A rotation in a 2D plane about the origin can be represented using rotation matrices or geometric reasoning.

- A **90 degrees counterclockwise rotation** transforms coordinates (x, y) to (-y, x).
- A **90 degrees clockwise rotation** transforms (x, y) to (y, -x).
- A **180 degrees rotation** transforms (x, y) to (-x, -y).
- A **270 degrees counterclockwise rotation** or a **90 degrees clockwise rotation** switches the transformations.

3. **Translating the coordinates**:
Let's check how point P(−4, 10) would be positioned if we rotated it through the various degrees:

- **90 degrees counterclockwise**:
\[
(−4, 10) \rightarrow (−10, −4)
\]
This matches the point P′.

This means the transformation from P to P′ is indeed a 90-degree counterclockwise rotation.

Thus, the degree of rotation is **90 degrees counterclockwise**.

Therefore, the correct choice is:

**90 degrees (counterclockwise)**.