Question

find the length of the third side. if necessary, round to the nearest tenth.

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem 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 we are finding the hypotenuse) or \(b^{2}=c^{2}-a^{2}\) (if we are finding a leg). Here, the two given sides are the legs (\(a = 6\), \(b\) is unknown, and \(c = 8\)? Wait, no, wait. Wait, in the right - triangle, the right - angle is between the side of length 6 and the unknown side, and the side of length 8 is the hypotenuse? Wait, no, let's check again. Wait, the right - angle is at the vertex where the side of length 6 and the unknown side meet. So the two legs are 6 and \(x\) (let the unknown side be \(x\)), and the hypotenuse is 8? Wait, no, that can't be, because \(6^{2}+x^{2}=8^{2}\) would give \(x^{2}=64 - 36=28\), \(x=\sqrt{28}\approx5.3\), but that would mean the hypotenuse is 8, but 8 is longer than 6, but if the hypotenuse is 8, then the other leg is shorter than 8. Wait, no, maybe I got the hypotenuse wrong. Wait, maybe the side of length 8 is a leg? Wait, no, in a right - triangle, the hypotenuse is the longest side. Since 8>6, so 8 is either the hypotenuse or a leg. If 8 is a leg, then the hypotenuse \(c\) would satisfy \(c^{2}=6^{2}+8^{2}\). Let's calculate that.

Step2: Apply the Pythagorean theorem

Let the unknown side be \(c\) (hypotenuse). Then by Pythagorean theorem \(c^{2}=6^{2}+8^{2}\)
\(6^{2}=36\), \(8^{2}=64\)
So \(c^{2}=36 + 64=100\)
Then \(c=\sqrt{100}=10\)

Wait, I made a mistake earlier. The side of length 8 is a leg, not the hypotenuse. Because if the right - angle is between 6 and the unknown side, then 6 and 8 are the legs, and the unknown side is the hypotenuse. So that's the correct approach.

Answer:

The length of the third side is 10.