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The problem involves a rectangle \( ABCD \), which represents a park, and a circle in the center of the rectangle. The dimensions of the rectangle are 50 meters by 80 meters, and the circle has a diameter of 15 meters.

You are asked to calculate the shortest distance from point \( A \) to point \( C \), following the paths along the edges of the rectangle and avoiding the circular path.

### Steps to solve:

1. **Diagonal of the Rectangle (Ignoring the Circle)**:
   The shortest path from \( A \) to \( C \), assuming no obstacles, would be along the diagonal of the rectangle. This diagonal can be calculated using the Pythagorean theorem:
   
   \[
   \text{Diagonal} = \sqrt{(80^2 + 50^2)}
   \]
   This will give us the direct diagonal from \( A \) to \( C \).

2. **Circle Adjustment**:
   Since the circle is located centrally with a diameter of 15 meters, we need to check how much this circle affects the diagonal. The circle's radius is \( 7.5 \) meters, and it is likely that the shortest path must go around the circle, adding some extra distance to the diagonal.

### Let me calculate the exact values and adjustment due to the circle.The direct diagonal distance from point \( A \) to point \( C \), without considering the circle, is approximately \( 94.34 \) meters.

Now, since there is a circular path in the middle with a diameter of 15 meters, the path from \( A \) to \( C \) must go around the circle. To determine the exact adjustment for this detour, the path around the circle adds extra distance. I'll calculate the best detour path that minimizes the added distance.To account for the circular path in the center, the shortest way around it is to follow a path that just skims around the circle's edge. Given that the radius of the circle is 7.5 meters, the extra distance is small but significant.

The minimal detour adds roughly twice the radius of the circle (15 meters), as this will account for moving around the circle. 

Thus, the adjusted shortest distance from \( A \) to \( C \) would be:

\[
\text{Adjusted Distance} \approx 94.34 + 15 = 109.34 \, \text{meters}
\]

### Final Answer:
The shortest distance from \( A \) to \( C \), following the paths around the circle, is approximately **109.34 meters**.

Would you like more details on this, or do you have further questions?

### Here are 5 related questions:
1. How would the distance change if the circle had a smaller diameter?
2. Can the Pythagorean theorem be used for any polygonal shapes?
3. What would the distance be if you could pass through the circular path instead of going around?
4. How can the perimeter of the circular path be calculated?
5. What happens to the path if the circle is not centered?

### Tip:
Always double-check units in geometry problems to ensure that no conversions are needed during calculations!

ABCD is a rectangle that represents a park. The lines show all the paths in the park. The circular path is in the center of the rectangle and has a diameter of 15 m. Calculate the shortest distance from A to C across the park, using only the lines shown.

Learn how to calculate the shortest distance from point A to point C across a rectangular park with a central circular obstacle. This problem involves using the Pythagorean theorem to find the diagonal and adjusting the path to account for the circle in the center. Suitable for students in Grades 8-10.

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