1. **State the problem:** We need to solve for $x$ in a right triangle with sides 10, 5, and $x$, and a smaller right triangle inside with one leg 4 and an unknown vertical side. 2. **Identify the triangles and relationships:** The large triangle has sides 10, 5, and $x$. The smaller triangle shares the right angle on the bottom side, with one leg 4 and the other leg unknown. 3. **Use the Pythagorean theorem:** For the large triangle, if the sides are 10 (hypotenuse), 5 (one leg), and $x$ (other leg), then $$10^2 = 5^2 + x^2$$ 4. **Calculate $x^2$:** $$100 = 25 + x^2$$ 5. **Isolate $x^2$:** $$x^2 = 100 - 25 = 75$$ 6. **Simplify $x$:** $$x = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ 7. **Answer:** The length $x$ in simplest radical form is $5\sqrt{3}$.