https://file.aiquickdraw.com/mathbot/file/1727088185317w8dhf5ws.jpg

To solve this problem, we need to rotate the point \((8, -3)\) 180 degrees counterclockwise about the center \((7, -1)\).

### Steps to solve:

1. **Translate the points**: First, we translate the center \((7, -1)\) to the origin by subtracting the coordinates of the center from both the point and the center.
   \[
   \text{Translation:} \quad (x, y) \rightarrow (x - 7, y + 1)
   \]
   Applying this to the point \((8, -3)\):
   \[
   (8 - 7, -3 + 1) = (1, -2)
   \]

2. **Rotate 180 degrees**: Rotating 180 degrees about the origin negates both the x and y coordinates:
   \[
   (x, y) \rightarrow (-x, -y)
   \]
   Applying this to \((1, -2)\):
   \[
   (-1, 2)
   \]

3. **Translate back**: Finally, translate the point back to the original center by adding the coordinates of the center to the rotated point:
   \[
   (-1 + 7, 2 - 1) = (6, 1)
   \]

The coordinates of the image after a 180-degree rotation are \((6, 1)\), which corresponds to **option B**.

Would you like more details on the process or have any questions?

### Related questions:
1. How do you rotate a point 90 degrees clockwise about an arbitrary center?
2. What happens when you rotate a point 270 degrees counterclockwise about the origin?
3. Can you explain how to find the midpoint between two points?
4. How does reflection across a line differ from rotation?
5. What are the coordinates of a point reflected across the y-axis?

**Tip**: When rotating points around a center that isn't the origin, translating the points to and from the origin simplifies calculations.

The point (8, -3) is rotated 180 degrees counterclockwise using center (7, -1). What are the coordinates of the image?

This problem demonstrates how to rotate the point (8, -3) by 180 degrees counterclockwise around the center (7, -1). Learn step-by-step translations and rotations of points in coordinate geometry. Suitable for Grades 9-11.

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