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 the sum of squares

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

Step3: Solve for \(c\)

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

Answer:

\(3\sqrt{2}\)