Question

The point P (−1,2) is rotated to become P′ (2,1). Describe the rotation by degree and direction.(1 point)Responses−270° rotationnegative 270 degrees rotation90° rotation90 degrees rotation−90° rotationnegative 90 degrees rotation180° rotation

Answer 1

To determine the rotation that takes point P (−1, 2) to point P′ (2, 1), we can analyze the transformation geometrically.

1. **Find the center of rotation**: A convenient center of rotation is often the origin (0, 0).

2. **Calculate the position before and after rotation**:
- Point P (−1, 2) relative to the origin is at (−1, 2).
- Point P′ (2, 1) relative to the origin is at (2, 1).

3. **Check the rotation**: To find the angle of rotation, we can use the following method:
- Determine the angle of the line connecting the origin to each point before and after the rotation using the inverse tangent function.
- For P (−1, 2):
- The slope is \( \frac{2 - 0}{-1 - 0} = -2 \).
- The angle θ1 = arctan(-2) results in an angle measured counterclockwise.
- For P' (2, 1):
- The slope is \( \frac{1 - 0}{2 - 0} = 0.5 \).
- The angle θ2 = arctan(0.5).

4. **Calculate the angle difference**: The difference in angles will give the rotation. You can find this by calculating both angles and determining their difference.

5. **Identify the degrees**: After evaluation, the clear rotation that takes P to P' while working through the necessary components leads us to find that P has actually moved to the right and down, indicating a possible clockwise rotation.

In conclusion, since a counterclockwise rotation of -90 degrees also has the same effect as a clockwise rotation of 270 degrees, the rotation we are looking for is:

**−90° rotation** (negative 90 degrees rotation).