https://file.aiquickdraw.com/mathbot/file/1730232614616ovdtv118.jpg

To find the coordinates of \( A' \) after a 180° counterclockwise rotation of point \( A(6, 4) \) about the point \( C(-1, 3) \), we can use the following steps:

### Steps
1. **Find the vector from \( C \) to \( A \)**:
   \[
   \text{Vector } CA = (6 - (-1), 4 - 3) = (7, 1)
   \]

2. **Rotate this vector by 180°**:
   A 180° rotation reverses the direction of the vector, making \( (7, 1) \) become \( (-7, -1) \).

3. **Apply this rotated vector to point \( C \)**:
   Add the rotated vector to the coordinates of \( C \):
   \[
   A' = (-1 + (-7), 3 + (-1)) = (-8, 2)
   \]

### Conclusion
The coordinates of \( A' \) are \( (-8, 2) \).

### Answer
The correct choice is:
**D. \( A'(-8, 2) \)**

Would you like further details or have any questions?

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### Related Questions
1. How would the coordinates change if the rotation was 90° instead of 180°?
2. How can you determine the coordinates of a point after a 270° rotation around a different center point?
3. What is the effect of rotating a point 180° around the origin?
4. How do you generalize the formula for rotating any point around a given center point by any angle?
5. What are the properties of transformations that preserve distance, such as rotation and reflection?

### Tip
When performing a rotation, visualizing the shift in coordinates around the pivot point can help ensure accuracy, especially when working with transformations on the coordinate plane.

Jenny’s art project requires her to rotate point A(6, 4) 180° counterclockwise about the point C(−1, 3). What are the coordinates of A'?

Solve a geometry problem involving a 180° rotation of point A(6, 4) about point C(−1, 3). Learn how to find the new coordinates using rotation formulas, suitable for students in Grades 9-10.

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