Question

a point is rotated about the origin. it's coordinates are P(-4,10) and P'(-10,-4). determine the degree of rotation and direction by applying rotation mapping.
A. 270 degrees (counterclockwise)
B. -90 degrees (clockwise)
C. 90 degrees (counterclockwise)
D. -180 degrees (clockwise) or 180 degrees (counterclockwise)

Answer 1

To determine the degree of rotation and direction when a point P is rotated about the origin to a new position P', we can apply rotation mapping.

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

To find the degree and direction of rotation, we can use the rotation formulas:

1. A point \( (x, y) \) rotated by an angle \( \theta \) around the origin becomes:
\[
(x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta))
\]

2. We need to find an angle \( \theta \) such that:
\[
-10 = -4 \cos(\theta) - 10 \sin(\theta)
\]
\[
-4 = -4 \sin(\theta) + 10 \cos(\theta)
\]

To simplify, let's check existing rotation angles and see which results in the correct coordinates after applying the transformations.

1. For **90 degrees counterclockwise** (which is \(\theta = 90\)):
- New coordinates: \( (y, -x) \) results in \( (10, 4) \) (not correct).

2. For **180 degrees** (which is \(\theta = 180\)):
- New coordinates: \( (-x, -y) \) results in \( (4, -10) \) (not correct).

3. For **270 degrees counterclockwise** (or **-90 degrees clockwise**):
- New coordinates: \( (-y, x) \) results in \( (-10, -4) \) (correct).

Thus, the point \( P(-4, 10) \) rotates to \( P'(-10, -4) \) through a rotation of **270 degrees counterclockwise** or equivalently **-90 degrees clockwise**.

So the correct answers would be:
- **A. 270 degrees (counterclockwise)**
- **B. -90 degrees (clockwise)**

If you need to choose one option from the list provided, you can select either **A or B**.