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 hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\). Let the hypotenuse \(c = 3\sqrt{5}\) and one leg \(b = 9\). We need to find the other leg \(a\). So, \(a^{2}=c^{2}-b^{2}\).
Substitute \(c = 3\sqrt{5}\) and \(b = 9\) into the formula: \(a^{2}=(3\sqrt{5})^{2}-9^{2}\).

Step2: Calculate \((3\sqrt{5})^{2}\) and \(9^{2}\)

Using the rule \((ab)^{n}=a^{n}b^{n}\), \((3\sqrt{5})^{2}=3^{2}\times(\sqrt{5})^{2}=9\times5 = 45\), and \(9^{2}=81\).
Then \(a^{2}=45 - 81=- 36\) (This is wrong, we made a mistake in step1. Let's correct it. The hypotenuse should be the longest side. Since \(3\sqrt{5}=\sqrt{9\times5}=\sqrt{45}\approx6.7\lt9\), so the leg is \(3\sqrt{5}\) and the other leg is \(a\), and the hypotenuse \(c = 9\)).
Now, using \(c^{2}=a^{2}+b^{2}\), where \(c = 9\), \(b = 3\sqrt{5}\). Then \(a^{2}=c^{2}-b^{2}\).
Substitute \(c = 9\) and \(b = 3\sqrt{5}\) into the formula: \(a^{2}=9^{2}-(3\sqrt{5})^{2}\).
Calculate \(9^{2}=81\) and \((3\sqrt{5})^{2}=9\times5 = 45\).
So \(a^{2}=81 - 45=36\).

Step3: Solve for \(a\)

Take the square root of both sides: \(a=\sqrt{36}\).

Answer:

\(6\)