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)Responses90 degrees (counterclockwise)90 degrees (counterclockwise)270 degrees (counterclockwise)270 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 for the point P(−4,10) to the point P′(−10,−4), we can use the rotation mapping formulas.

1. **Identify coordinates**:
- Original point \( P(x, y) = (-4, 10) \)
- Rotated point \( P'(x', y') = (-10, -4) \)

2. **Check the possible rotation formulas**. The rotation formulas involving degrees are:
- For 90 degrees counterclockwise:
\[
(x', y') = (-y, x)
\]
- For 180 degrees (both directions):
\[
(x', y') = (-x, -y)
\]
- For 270 degrees counterclockwise (or 90 degrees clockwise):
\[
(x', y') = (y, -x)
\]

3. **Test a 90-degree counterclockwise rotation**:
\[
P \text{ becomes } (-y, x) = (-10, -4)
\]
This does not match P′.

4. **Test a 180-degree rotation**:
\[
P \text{ becomes } (-x, -y) = (4, -10)
\]
This does not match P′.

5. **Test a 270-degree counterclockwise rotation** (or 90 degrees clockwise):
\[
P \text{ becomes } (y, -x) = (10, 4)
\]
This does not match P′ as well.

6. **Test a 90-degree clockwise rotation** (which is the same as 270 degrees counterclockwise):
\[
P \text{ becomes } (-y, x) = (-10, -4)
\]
This does match P′.

Thus, the point \( P(−4, 10) \) is rotated to \( P' (−10, −4) \) through a **90 degrees clockwise rotation**.

So the answer is: **−90 degrees (clockwise)**.