The diagram shows a set of similar triangles. These two triangles are similar because they share the same angles (both are isosceles triangles).

Step-by-step solution:

1. Since the triangles are similar, the ratios of their corresponding sides are equal. We can set up a proportion based on this similarity:

   $\frac{x}{32} = \frac{56}{20}$

   The first ratio corresponds to the sides of the larger triangle, and the second ratio corresponds to the sides of the smaller triangle.

2. Solve the proportion:

   $x \cdot 20 = 56 \cdot 32$

   Simplify the right-hand side:

   $56 \cdot 32 = 1792$

   Now divide both sides by 20:

   $x = \frac{1792}{20} = 89.6$

Thus, the value of $x$ is 89.6 feet.

Would you like more details or have any questions?

Here are some related questions to further explore the topic:

1. What are the properties of similar triangles, and how can they help in problem-solving?
2. How do you determine the proportionality between different triangles in geometric figures?
3. What other real-world scenarios involve similar triangles or proportional relationships?
4. Can this method of solving proportions be applied to other types of geometric figures?
5. How would the solution change if the triangles were not isosceles?

Tip: Always check if the triangles are similar by verifying angle relationships or side ratios before setting up proportions.