This problem involves two similar triangles within a right triangle. The smaller triangle has sides of length 4 (adjacent side) and $x$ (hypotenuse), while the larger triangle has sides of length 10 (adjacent side) and 25 (hypotenuse). We can use the property of similar triangles to find the value of $x$.

Step-by-Step Solution:

1. Since the triangles are similar, their corresponding sides are proportional. This gives us the ratio: $\frac{4}{10} = \frac{x}{25}$

2. Cross-multiply to solve for $x$: $4 \times 25 = 10 \times x$ $100 = 10x$

3. Divide both sides by 10 to isolate $x$: $x = \frac{100}{10} = 10$

Thus, the value of $x$ is $\boxed{10}$.

Would you like further clarification on the method used? Here are five related questions to deepen your understanding:

1. What is the concept of similar triangles, and how can it be used to solve for unknown sides?
2. How do you identify corresponding sides in similar triangles?
3. Can you explain why the two triangles in this diagram are similar?
4. What happens if the triangles are not similar? Can you still solve for $x$?
5. How does cross-multiplication work in solving proportion equations?

Tip: In geometry, always look for similar triangles within complex shapes—they often provide a simpler way to find missing dimensions using proportional relationships.