question find the length of the third side. if necessary, write in simplest radical form. answer attempt 1…

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with sides $a$, $b$ and hypotenuse $c$, $a^{2}+b^{2}=c^{2}$. Let the two given sides be $a = 3\sqrt{3}$ and $b = 6$. We need to find the third side. There are two cases: if the third side is the hypotenuse $c$, then $c^{2}=(3\sqrt{3})^{2}+6^{2}$; if the third side is a non - hypotenuse side, say $a$, then $6^{2}=a^{2}+(3\sqrt{3})^{2}$. First, assume the third side is the hypotenuse. [c^{2}=(3\sqrt{3})^{2}+6^{2}] [c^{2}=3^{2}\times(\sqrt{3})^{2}+36] [c^{2}=9\times3 + 36] [c^{2}=27+36] [c^{2}=63] [c=\sqrt{63}=\sqrt{9\times7}=3\sqrt{7}] Now assume the third side is a non - hypotenuse side. Let the hypotenuse be 6. Then [6^{2}=a^{2}+(3\sqrt{3})^{2}] [36=a^{2}+27] [a^{2}=36 - 27=9] [a = 3] Since the hypotenuse is the longest side in a right - triangle and $6>3\sqrt{3}$, the third side we want (non - hypotenuse) is 3.

Answer:

3