Question

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

Explanation:

Step1: Use the Pythagorean theorem for the upper - right triangle

Let's first find the length of the common side of the two right - angled triangles. For the upper - right triangle with sides 4 and 8, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, the length of the hypotenuse of the upper - right triangle (which is also a side of the lower right - angled triangle) is $\sqrt{4^{2}+8^{2}}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}$.

Step2: Use the Pythagorean theorem for the lower - right triangle

For the lower right - angled triangle with sides 3 and the hypotenuse we just found ($4\sqrt{5}$), and the unknown side $x$. According to the Pythagorean theorem $x=\sqrt{(4\sqrt{5})^{2}-3^{2}}$. First, $(4\sqrt{5})^{2}=16\times5 = 80$ and $3^{2}=9$. Then $x=\sqrt{80 - 9}=\sqrt{71}$.

Answer:

$\sqrt{71}$