question\nfind the length of the third side. if necessary, write in simplest radical form.\nanswer attempt 1…

Explanation:

Step1: Identify the hypotenuse

Since it's a right - triangle, assume the hypotenuse is the longest side. We need to check if 9 or $3\sqrt{5}$ is the hypotenuse. Calculate $(3\sqrt{5})^2=3^2\times(\sqrt{5})^2 = 45$ and $9^2 = 81$. Since $81>45$, 9 is the hypotenuse.

Step2: Apply the Pythagorean theorem

Let the unknown side be $x$. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 9$ and $a = 3\sqrt{5}$. So $x^{2}+(3\sqrt{5})^{2}=9^{2}$, which simplifies to $x^{2}+45 = 81$.

Step3: Solve for $x$

Subtract 45 from both sides: $x^{2}=81 - 45=36$. Then take the square root of both sides. Since $x$ represents the length of a side of a triangle, we take the positive square - root. So $x=\sqrt{36}=6$.

Answer:

6