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To solve this problem, we need to use the property of a centroid in a triangle. The centroid (F) of a triangle divides each median into a ratio of 2:1, where the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

### Given:
- The fountain \( F \) is located at the centroid of the park's triangular shape.
- Distance from the centroid \( F \) to the gift shop (point \( C \)) is 86 feet.

### Step-by-step Solution:

1. **Understand the Problem Setup:**
   - The triangle is labeled with vertices \( A, B, \) and \( C \) representing the Entrance A, Petting Zoo, and Gift Shop, respectively.
   - \( F \) is the centroid of the triangle.
   - We need to find the length of the side walk from the Fountain \( F \) to Entrance \( A \).

2. **Using the Centroid Property:**
   - The centroid \( F \) divides the median from any vertex to the opposite side in a 2:1 ratio.
   - If the distance from \( F \) to \( C \) (the midpoint of side \( AB \)) is 86 feet, the full median length from vertex \( A \) to \( C \) (along the line \( AF \)) is three times that, because \( FC:FA = 1:2 \).

3. **Calculate the Full Length of Median \( AF \):**
   \[
   AF = 3 \times FC = 3 \times 86 = 258 \, \text{feet}.
   \]

4. **Determine the Required Distance \( FA \):**
   - Since \( F \) is 1/3 of the way along the median from vertex \( A \), and the remaining 2/3 is \( FA \), we use the ratio:
   \[
   FA = \frac{2}{3} \times 258 = 172 \, \text{feet}.
   \]

### Conclusion:

The length of the sidewalk from the Fountain \( F \) to Entrance \( A \) is **172 feet**.

Would you like more details on this explanation or have any questions?

### Further Questions:
1. What is the formula for finding the centroid of a triangle given the vertices' coordinates?
2. How can the centroid property be applied to solve real-world problems involving medians?
3. How would the calculation change if the triangle were not evenly spaced?
4. Can the centroid be located outside the triangle? Under what circumstances?
5. What are some applications of the centroid concept in physics?

### Tip:
The centroid is always located inside the triangle and can be used as a balancing point. This property is useful in various fields like engineering, physics, and design.

What is the length of the side walk from the Fountain to Entrance A?

Solve a real-world geometry problem using the centroid theorem of a triangle. In this example, the distance from a park's fountain (located at the centroid) to Entrance A is found using properties of the centroid, dividing the median in a 2:1 ratio.

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