https://file.aiquickdraw.com/mathbot/file/17260293131523w8nql8y.jpg

To solve the problem shown in the image, which involves rotating a triangle on the coordinate plane about the origin \((0,0)\), let's break it down step by step.

### Given Points
- \( A(-2, 2) \)
- \( B(-3, 1) \)
- \( C(0, 0) \)

### Rotation Scenarios
1. **90° Counterclockwise Rotation**:
   - The rotation rule for 90° counterclockwise about the origin is \((x, y) \rightarrow (-y, x)\).
   - \( A(-2, 2) \) becomes \( A'(-2, -2) \).
   - \( B(-3, 1) \) becomes \( B'(-1, -3) \).
   - \( C(0, 0) \) remains \( C'(0, 0) \).

2. **180° Rotation (Clockwise or Counterclockwise)**:
   - The rotation rule for 180° about the origin is \((x, y) \rightarrow (-x, -y)\).
   - \( A(-2, 2) \) becomes \( A'(2, -2) \).
   - \( B(-3, 1) \) becomes \( B'(3, -1) \).
   - \( C(0, 0) \) remains \( C'(0, 0) \).

3. **270° Counterclockwise Rotation (Equivalent to 90° Clockwise)**:
   - The rotation rule for 270° counterclockwise about the origin is \((x, y) \rightarrow (y, -x)\).
   - \( A(-2, 2) \) becomes \( A'(2, 2) \).
   - \( B(-3, 1) \) becomes \( B'(1, 3) \).
   - \( C(0, 0) \) remains \( C'(0, 0) \).

### Explanation
- If you choose **90° counterclockwise**, the triangle rotates in the positive direction (counterclockwise), resulting in new coordinates where each point’s x and y values are swapped and the y-coordinate is negated.
- For **180° rotation**, both x and y coordinates of each point are negated, effectively flipping the triangle around the origin.
- For **270° counterclockwise (or 90° clockwise)**, the coordinates swap similarly to the 90° counterclockwise rotation, but the x-coordinate is negated instead of the y-coordinate.

### Example of Writing About What You Did:
- "I rotated the pre-image triangle 90° counterclockwise about the origin. Using the rotation rule \((x, y) \rightarrow (-y, x)\), the vertices of the triangle moved as follows: point \(A(-2, 2)\) rotated to \(A'(-2, -2)\), point \(B(-3, 1)\) rotated to \(B'(-1, -3)\), and point \(C(0, 0)\) remained unchanged."

Would you like to see more detailed steps or have any further questions?

### Related Questions:
1. What is the effect of a 360° rotation on any point in a plane?
2. How does rotating a shape affect its orientation and size?
3. Can you describe the steps to rotate a shape 45° around a point that is not the origin?
4. How would the rotation rules change if the rotation center was \((1, 1)\) instead of \((0, 0)\)?
5. How do you determine the center of rotation given an original shape and its image?

### Tip:
When rotating points on a coordinate plane, always pay attention to the signs of the coordinates after applying the rotation rule, as this will determine the correct quadrant placement of the rotated image.

Rotate the given triangle on the coordinate plane about the origin (0,0). Decide the direction and degree of rotation (90°, 180°, 270°, etc.) and describe the transformation process.

Explore how to rotate a triangle around the origin on a coordinate plane. Learn how to apply rotation formulas for 90°, 180°, and 270° degrees, including counterclockwise and clockwise rotations. Suitable for Grades 8-10 students.

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