Question

The point P (4,2) is rotate to become P'(-2,4). Describe the rotation by degree and direction.

Answer 1

To find the rotation that takes the point \( P(4, 2) \) to the point \( P'(-2, 4) \), we can use the following approach:

1. **Determine the center of rotation**:
This can be a bit complex, but we can find the rotation angle without needing to identify the exact center first.

2. **Calculate the angle of rotation**:
We can visualize the points in the Cartesian plane and determine the required angle of rotation. When we rotate a point around the origin, the angle of rotation can be calculated using the coordinates of the points.

The original point \( P(4, 2) \) can be expressed in polar coordinates, where:
- \( r = \sqrt{x^2 + y^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \)
- \( \theta = \tan^{-1}\left( \frac{y}{x} \right) = \tan^{-1}\left( \frac{2}{4} \right) = \tan^{-1}\left( \frac{1}{2} \right) \)

For the new point \( P'(-2, 4) \):
- \( r' = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \) (same distance from the origin)
- \( \theta' = \tan^{-1}\left( \frac{4}{-2} \right) = \tan^{-1}(-2) + 180^\circ\).

3. **Determine the angle**:
The angle \( P \) makes with the positive x-axis in standard position is about:
\[
\theta = \tan^{-1}(0.5) \approx 26.57^\circ
\]

The angle \( P' \) makes with the positive x-axis:
\[
\theta' = \tan^{-1}\left(-2\right) + 180^\circ \approx -63.43^\circ + 180^\circ \approx 116.57^\circ
\]

To find the angle of rotation, calculate:
\[
\text{Rotation angle} = \theta' - \theta \approx 116.57^\circ - 26.57^\circ = 90^\circ.
\]

4. **Direction of the rotation**:
Since the counter-clockwise direction is considered positive, the result indicates that the rotation is counter-clockwise.

Thus, the point \( P(4, 2) \) is rotated **90 degrees counterclockwise** to become \( P'(-2, 4) \).