To solve for $x$ in this diagram, we can approach the problem as a multi-step application of the Pythagorean Theorem.

Step-by-Step Solution

1. Identify Right Triangles:
   The diagram contains two right triangles. The smaller triangle has sides of 2 and 5, while the larger triangle has hypotenuse $x$ and leg lengths of 7 and the hypotenuse of the smaller triangle (which we need to find).

2. Calculate Hypotenuse of the Smaller Triangle:
   For the smaller triangle with legs 2 and 5, let $c$ be the hypotenuse: $c = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}$

3. Use the Larger Triangle:
   Now we can apply the Pythagorean Theorem to the larger triangle with sides 7 and $\sqrt{29}$, and hypotenuse $x$: $x = \sqrt{7^2 + (\sqrt{29})^2} = \sqrt{49 + 29} = \sqrt{78}$

4. Final Answer in Simplest Radical Form:
   So, the answer is: $x = \sqrt{78}$

If you need further explanation or have any questions, feel free to ask!

Here are five related questions to expand your understanding:

1. How do we simplify square roots, like $\sqrt{78}$?
2. What would change if the dimensions of the smaller triangle were different?
3. How does the Pythagorean Theorem apply to non-right triangles?
4. What are some real-world applications of the Pythagorean Theorem?
5. How would you solve this if only one triangle’s side lengths were given?

Tip: When solving multi-step problems, break them down into individual triangles to apply the Pythagorean Theorem accurately.