Question

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

Explanation:

Step1: Find the length of the hypotenuse of the upper - right triangle

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) for the right - triangle with sides 4 and 8. Let \(a = 4\) and \(b = 8\), then \(c=\sqrt{4^{2}+8^{2}}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\).

Step2: Find the length of the hypotenuse of the lower - right triangle

Use the Pythagorean theorem for the right - triangle with sides 3 and \(x\). The hypotenuse of the lower - right triangle is the same as the hypotenuse of the upper - right triangle. So, by the Pythagorean theorem \(3^{2}+x^{2}=(4\sqrt{5})^{2}\).

Step3: Solve the equation for \(x\)

Expand the equation: \(9+x^{2}=80\). Then, subtract 9 from both sides: \(x^{2}=80 - 9=71\). Take the square root of both sides: \(x=\sqrt{71}\).

Answer:

\(x = \sqrt{71}\)