1. **State the problem:** We have a large right triangle divided into two smaller right triangles by a height line. The base is split into segments of lengths 5 and 4, the left side of the large triangle is 10, and the right side is $x$. We need to solve for $x$. 2. **Identify the triangles and apply the Pythagorean theorem:** The height line creates two right triangles. Let the height be $h$. For the left triangle with base 5 and hypotenuse 10: $$10^2 = h^2 + 5^2$$ $$100 = h^2 + 25$$ $$h^2 = 100 - 25 = 75$$ $$h = \sqrt{75} = 5\sqrt{3}$$ 3. **Use the height $h$ to find $x$ in the right triangle:** The right triangle has base 4, height $h = 5\sqrt{3}$, and hypotenuse $x$. Apply the Pythagorean theorem: $$x^2 = h^2 + 4^2$$ $$x^2 = 75 + 16 = 91$$ 4. **Simplify the radical:** $$x = \sqrt{91}$$ Since 91 factors as $7 \times 13$ and neither is a perfect square, $\sqrt{91}$ is already in simplest radical form. **Final answer:** $$x = \sqrt{91}$$