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To find the coordinates of point \( A' \) after rotating point \( A(6, 4) \) 180° counterclockwise around point \( C(-1, 3) \), we can follow these steps:

1. **Determine the translation vector** from \( C \) to \( A \).
   - From \( C(-1, 3) \) to \( A(6, 4) \), the change in \( x \) is \( 6 - (-1) = 7 \).
   - The change in \( y \) is \( 4 - 3 = 1 \).

   So, the vector from \( C \) to \( A \) is \( (7, 1) \).

2. **Apply a 180° rotation** to the vector \( (7, 1) \).
   - A 180° rotation changes the vector \( (7, 1) \) to \( (-7, -1) \).

3. **Translate from point \( C \) using the rotated vector**.
   - Start at \( C(-1, 3) \) and add the rotated vector \( (-7, -1) \).
   - \( x \)-coordinate: \( -1 + (-7) = -8 \).
   - \( y \)-coordinate: \( 3 + (-1) = 2 \).

   Therefore, the coordinates of \( A' \) are \( (-8, 2) \).

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

Would you like more details or have any questions?

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Here are five related questions for further practice:

1. How do you perform a 90° counterclockwise rotation around a point?
2. What are the effects of a 270° rotation on coordinates?
3. How can you calculate the distance between two points in the coordinate plane?
4. How would the coordinates change if the rotation was clockwise instead of counterclockwise?
5. How do translations affect rotations in the coordinate plane?

**Tip:** When performing rotations, always remember that a 180° rotation is the same as reflecting the point over the center of rotation.

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'?

Learn how to find the new coordinates of point A(6, 4) after a 180° counterclockwise rotation around point C(-1, 3). This problem includes essential geometry concepts such as transformations and rotation rules, suitable for Grade 8 students.

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