Question

find the length of the third side. if necessary, write in simplest radical form. answer attempt 2 out of 2

Explanation:

Step1: Apply Pythagorean theorem

Let the sides of the right - triangle be $a = 3$, $b$ (the unknown side), and $c=3\sqrt{3}$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ for a right - triangle with hypotenuse $c$. In this case, we need to find $b$, so $b=\sqrt{c^{2}-a^{2}}$.

Step2: Substitute the values of $a$ and $c$

Substitute $a = 3$ and $c = 3\sqrt{3}$ into the formula. First, calculate $c^{2}=(3\sqrt{3})^{2}=3^{2}\times(\sqrt{3})^{2}=9\times3 = 27$ and $a^{2}=3^{2}=9$. Then $b=\sqrt{27 - 9}$.

Step3: Simplify the expression

$b=\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}$.

Answer:

$3\sqrt{2}$