Question

find the length of the third side. if necessary, round to the nearest tenth.
image of a right triangle with one leg labeled 8, hypotenuse labeled 10, and the right angle between the leg labeled 8 and the other leg (the third side to find).

Explanation:

Step1: Identify triangle type

It's a right - triangle, so use Pythagorean theorem: \(a^{2}+b^{2}=c^{2}\), where \(c = 10\) (hypotenuse), \(b = 8\) (one leg), find \(a\) (the other leg).

Step2: Apply Pythagorean theorem

Rearrange formula: \(a=\sqrt{c^{2}-b^{2}}\). Substitute \(c = 10\), \(b = 8\): \(a=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6\)? Wait, no, wait. Wait, the right angle is at the vertex with the two legs? Wait, no, the hypotenuse is the longest side. Wait, if the sides are 8, 10, and the third side. Wait, maybe I mixed up. Wait, in a right - triangle, hypotenuse is the longest side. So if 10 is the hypotenuse, and 8 is one leg, then the other leg \(x\) satisfies \(x^{2}+8^{2}=10^{2}\). So \(x^{2}=100 - 64 = 36\), so \(x = 6\)? But wait, maybe the 10 is a leg? No, 10 is longer than 8, so 10 should be hypotenuse. Wait, but let's check again. The triangle has a right angle, so the two legs are the sides forming the right angle, and hypotenuse is opposite. So if the sides given are 8 (one leg), 10 (hypotenuse), then the third side (leg) is \(\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6\). Wait, but that seems too simple. Wait, maybe I misread the triangle. Wait, the triangle has sides 8, 10, and the third side. Let me confirm the Pythagorean theorem. For a right - triangle, \(c^{2}=a^{2}+b^{2}\), where \(c\) is hypotenuse. So if \(c = 10\), \(a = 8\), then \(b=\sqrt{c^{2}-a^{2}}=\sqrt{100 - 64}=\sqrt{36}=6\). Yes, that's correct.

Answer:

6