Question
The point P (−1,2) is rotated to become P′ (2,1) . Describe the rotation by degree and direction.(1 point)
Responses
−270° rotation
negative 270 degrees rotation
−90° rotation
negative 90 degrees rotation
90° rotation
90 degrees rotation
180° rotation
Responses
−270° rotation
negative 270 degrees rotation
−90° rotation
negative 90 degrees rotation
90° rotation
90 degrees rotation
180° rotation
Answers
GPT-4o mini
To determine the rotation that transforms the point \( P(-1, 2) \) to \( P'(2, 1) \), we can analyze the coordinates.
1. **Original Point**: \( P(-1, 2) \)
2. **New Point**: \( P'(2, 1) \)
### Steps to Find the Rotation:
- Look at the transformation of the coordinates.
The rotation can be calculated by analyzing the movement of the point in relation to the origin (0, 0).
### Visual Analysis:
- **Point P(-1, 2)** lies in the second quadrant (left and above the origin).
- **Point P'(2, 1)** lies in the first quadrant (right and above the origin).
The transformation indicates that the point has moved from quadrant II to quadrant I. To find the degree of rotation needed to achieve this, let's consider the coordinates.
### Calculation of the Rotation:
1. **Original Coordinates** \( (-1, 2) \)
2. **New Coordinates** \( (2, 1) \)
We can see that the point has effectively moved to the right (positive x-direction) and slightly down (decreased in y-value).
To find the angle:
1. The original angle \( P(-1, 2) \) with respect to the x-axis can be determined from the tangent (which would point diagonal to the origin).
2. The point \( P'(2, 1) \) has a tangent that can also indicate a similar angle.
### Testing Possible Rotations:
- **90° rotation counterclockwise** would change the point as follows:
- If you rotate \( P(-1, 2) \):
- New coordinates after a 90° counterclockwise rotation:
- Original x-coordinate becomes negative y-coordinate, and original y-coordinate becomes x-coordinate.
- Transforms to \( (-2, -1) \) which does not match.
- **-90° or 270° rotation** clockwise:
- P moves to \( (2, -1) \).
Though this does not match yet.
Through deduction by checking:
- A **-90° (or clockwise)** rotation would correctly transform:
- From \( (-1, 2) \) directly to \( (2, 1) \) which through transformation flows logically as:
### Conclusion
Thus the rotation required from \( P(-1, 2) \) to \( P'(2, 1) \) is indeed:
**Answer**: **−90° rotation (negative 90 degrees rotation)**.
1. **Original Point**: \( P(-1, 2) \)
2. **New Point**: \( P'(2, 1) \)
### Steps to Find the Rotation:
- Look at the transformation of the coordinates.
The rotation can be calculated by analyzing the movement of the point in relation to the origin (0, 0).
### Visual Analysis:
- **Point P(-1, 2)** lies in the second quadrant (left and above the origin).
- **Point P'(2, 1)** lies in the first quadrant (right and above the origin).
The transformation indicates that the point has moved from quadrant II to quadrant I. To find the degree of rotation needed to achieve this, let's consider the coordinates.
### Calculation of the Rotation:
1. **Original Coordinates** \( (-1, 2) \)
2. **New Coordinates** \( (2, 1) \)
We can see that the point has effectively moved to the right (positive x-direction) and slightly down (decreased in y-value).
To find the angle:
1. The original angle \( P(-1, 2) \) with respect to the x-axis can be determined from the tangent (which would point diagonal to the origin).
2. The point \( P'(2, 1) \) has a tangent that can also indicate a similar angle.
### Testing Possible Rotations:
- **90° rotation counterclockwise** would change the point as follows:
- If you rotate \( P(-1, 2) \):
- New coordinates after a 90° counterclockwise rotation:
- Original x-coordinate becomes negative y-coordinate, and original y-coordinate becomes x-coordinate.
- Transforms to \( (-2, -1) \) which does not match.
- **-90° or 270° rotation** clockwise:
- P moves to \( (2, -1) \).
Though this does not match yet.
Through deduction by checking:
- A **-90° (or clockwise)** rotation would correctly transform:
- From \( (-1, 2) \) directly to \( (2, 1) \) which through transformation flows logically as:
### Conclusion
Thus the rotation required from \( P(-1, 2) \) to \( P'(2, 1) \) is indeed:
**Answer**: **−90° rotation (negative 90 degrees rotation)**.