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To find the center of rotation for the triangles \( \triangle ABC \) and \( \triangle A'B'C' \) shown in the image, we can follow these steps:

1. **Identify corresponding points**: The points \( A \), \( B \), and \( C \) on \( \triangle ABC \) correspond to the points \( A' \), \( B' \), and \( C' \) on \( \triangle A'B'C' \) respectively.

2. **Draw the perpendicular bisectors**: 
   - Find the midpoints of the line segments \( AA' \), \( BB' \), and \( CC' \).
   - Draw the perpendicular bisectors of these segments. The point where at least two of these bisectors intersect is the center of rotation.

3. **Intersection Point**: The intersection of these perpendicular bisectors is the center of rotation.

Given the grid, I will determine the midpoints and slopes for perpendicular bisectors and find the intersection point manually based on the image. 

### Calculations:

- **Midpoint of \( AA' \)**: 
  - \( A = (2, 6) \) and \( A' = (0, -4) \).
  - Midpoint \( M_A = \left(\frac{2+0}{2}, \frac{6+(-4)}{2}\right) = (1, 1) \).

- **Midpoint of \( BB' \)**: 
  - \( B = (5, 7) \) and \( B' = (1, -3) \).
  - Midpoint \( M_B = \left(\frac{5+1}{2}, \frac{7+(-3)}{2}\right) = (3, 2) \).

- **Midpoint of \( CC' \)**:
  - \( C = (6, 4) \) and \( C' = (2, -6) \).
  - Midpoint \( M_C = \left(\frac{6+2}{2}, \frac{4+(-6)}{2}\right) = (4, -1) \).

Next, we need to find the slopes of the lines connecting the points and the slopes of the perpendicular bisectors.

- **Slope of \( AA' \)**: \( m = \frac{-4-6}{0-2} = \frac{-10}{-2} = 5 \). Perpendicular slope = \( -\frac{1}{5} \).
- **Slope of \( BB' \)**: \( m = \frac{-3-7}{1-5} = \frac{-10}{-4} = \frac{5}{2} \). Perpendicular slope = \( -\frac{2}{5} \).
- **Slope of \( CC' \)**: \( m = \frac{-6-4}{2-6} = \frac{-10}{-4

Suppose triangle A'B'C' is a rotation of triangle ABC around some point. Find the center of rotation.

Learn how to determine the center of rotation for a rotated triangle ABC using perpendicular bisectors. This problem involves key geometric concepts and formulas, making it suitable for students in Grades 8-10.

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