Question

find the length of the third side. if necessary, write in simplest radical form

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 3\) and \(b = 3\), and hypotenuse \(c\), the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\).
Substitute \(a = 3\) and \(b = 3\) into the formula: \(3^{2}+3^{2}=c^{2}\).

Step2: Calculate \(a^{2}+b^{2}\)

\(3^{2}+3^{2}=9 + 9=18\), so \(c^{2}=18\).

Step3: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{18}\).
Simplify \(\sqrt{18}=\sqrt{9\times2}\).
Since \(\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}\) and \(\sqrt{9} = 3\), then \(c = 3\sqrt{2}\).

Answer:

\(3\sqrt{2}\)