Question

the triangle below is isosceles. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Apply Pythagorean theorem

In an isosceles right - triangle, if the legs have length \(a = 3\), and the hypotenuse is \(x\). By the Pythagorean theorem \(a^{2}+a^{2}=x^{2}\), substituting \(a = 3\), we get \(3^{2}+3^{2}=x^{2}\), so \(9 + 9=x^{2}\), and \(x^{2}=18\).

Step2: Solve for \(x\)

Take the square - root of both sides: \(x=\sqrt{18}\). Simplify \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\).

Answer:

\(3\sqrt{2}\)