solve for x. leave your answer in simplest radical form.

Explanation:

Step1: Use the Pythagorean theorem for the large - triangle.

Let the height of the large triangle be $h$. First, find the height using the Pythagorean theorem for the left - hand right - triangle. If the hypotenuse of the left - hand right - triangle is $9$ and the base is $6$, then $h=\sqrt{9^{2}-6^{2}}=\sqrt{81 - 36}=\sqrt{45}=3\sqrt{5}$.

Step2: Use the Pythagorean theorem for the right - hand right - triangle.

In the right - hand right - triangle, one leg is $h = 3\sqrt{5}$ and the other leg is $4$. Then, by the Pythagorean theorem $x=\sqrt{(3\sqrt{5})^{2}+4^{2}}$. Calculate $(3\sqrt{5})^{2}=3^{2}\times(\sqrt{5})^{2}=9\times5 = 45$ and $4^{2}=16$. So $x=\sqrt{45 + 16}=\sqrt{61}$.

Answer:

$\sqrt{61}$