Question

question
find the length of the third side. if necessary, write in simplest radical form.
answer attempt 1 out of 2

Explanation:

Step1: Identify triangle type (right triangle)

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if \(c\) is the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if \(a\) is a leg and \(c\) is the hypotenuse). Here, we need to check which side is the hypotenuse. The hypotenuse is the longest side in a right - triangle. We have one side as \(6\) and another as \(3\sqrt{5}\approx3\times2.24 = 6.72\). Wait, \(3\sqrt{5}\approx6.72>6\), so \(3\sqrt{5}\) is the hypotenuse, and we need to find the other leg. Let the unknown leg be \(x\). Then by the Pythagorean theorem, \((3\sqrt{5})^{2}=6^{2}+x^{2}\) is wrong. Wait, no: if the right - angle is between the side of length \(6\) and the unknown side \(x\), then the hypotenuse is \(3\sqrt{5}\)? No, that can't be because \(3\sqrt{5}\approx6.72\) and \(6\) is less than that, but actually, let's re - check. Wait, maybe I got the hypotenuse wrong. Let's assume the two legs are \(x\) and \(6\), and the hypotenuse is \(3\sqrt{5}\)? No, because \(3\sqrt{5}\approx6.72\), and \(6.72^{2}=(3\sqrt{5})^{2}=9\times5 = 45\), and \(6^{2}=36\). If we assume the hypotenuse is \(6\), then the formula would be \(6^{2}=(3\sqrt{5})^{2}+x^{2}\)? No, that would give \(x^{2}=36 - 45=- 9\), which is impossible. So the hypotenuse must be \(3\sqrt{5}\)? No, that can't be. Wait, maybe I made a mistake. Wait, \(3\sqrt{5}\) is approximately \(6.708\), and \(6\) is less than that. Wait, no, the right - angle is marked, so the two sides forming the right - angle are the legs, and the side opposite the right - angle is the hypotenuse. So if the right - angle is between the side of length \(6\) and the unknown side \(x\), then the hypotenuse is \(3\sqrt{5}\)? But \(3\sqrt{5}\approx6.708>6\), so that is possible. Wait, no, the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the side opposite the right - angle). So if the two legs are \(a\) and \(b\), and hypotenuse is \(c\), then \(c\) must be longer than either \(a\) or \(b\). So if one leg is \(6\), and the hypotenuse is \(3\sqrt{5}\), then \(3\sqrt{5}\) must be longer than \(6\), but \(3\sqrt{5}\approx6.708>6\), so that is okay. Wait, no, let's do it correctly. Let's let the two legs be \(x\) and \(6\), and hypotenuse be \(3\sqrt{5}\). Then \(x^{2}+6^{2}=(3\sqrt{5})^{2}\). Then \(x^{2}+36 = 45\), so \(x^{2}=45 - 36 = 9\), so \(x = 3\). Wait, that makes sense.

Wait, let's re - derive. The Pythagorean theorem: for a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). We need to find the length of the third side. Let's assume that the side of length \(6\) and the unknown side \(x\) are the legs, and \(3\sqrt{5}\) is the hypotenuse. Then:

\(6^{2}+x^{2}=(3\sqrt{5})^{2}\)

Step2: Substitute values into Pythagorean theorem

First, calculate \((3\sqrt{5})^{2}\): \((3\sqrt{5})^{2}=3^{2}\times(\sqrt{5})^{2}=9\times5 = 45\)

And \(6^{2}=36\)

So the equation becomes \(36+x^{2}=45\)

Step3: Solve for \(x^{2}\)

Subtract \(36\) from both sides of the equation: \(x^{2}=45 - 36=9\)

Step4: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{9}=3\) (we take the positive square root because length can't be negative)

Answer:

\(3\)