To solve for $x$ in this problem, we can apply the Pythagorean theorem in both of the right triangles shown.

Step-by-Step Solution

1. Identify the triangles and apply the Pythagorean theorem.

   We have two right triangles:

   • The larger triangle with sides 8 (leg), $x$ (hypotenuse), and a shared segment (which we will calculate).
   • The smaller triangle with sides 3 (leg), 4 (hypotenuse), and the same shared segment.

2. Calculate the shared segment in the smaller triangle.

   Using the Pythagorean theorem on the smaller triangle: $\text{shared segment} = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7}$

3. Apply the Pythagorean theorem to the larger triangle.

   For the larger triangle: $x = \sqrt{8^2 + (\sqrt{7})^2} = \sqrt{64 + 7} = \sqrt{71}$

Final Answer

$x = \sqrt{71}$

Would you like additional details on this solution?

Here are 5 related questions to explore further:

1. How would the solution change if one of the triangle sides were different?
2. Can the Pythagorean theorem be applied in non-right triangles?
3. How do we determine which sides are legs and which is the hypotenuse in any triangle?
4. What is the geometric interpretation of $\sqrt{71}$?
5. Can we solve similar problems using trigonometric ratios instead of the Pythagorean theorem?

Tip: Always check that the triangle is a right triangle before applying the Pythagorean theorem.