Question

find the length of the third side. if necessary, round to the nearest tenth.
3
4

Explanation:

Step1: Identify triangle type

It's a right triangle, so use Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(a = 3\), \(b =?\), \(c = 4\) (wait, no—wait, the right angle is between the side of length 3 and the unknown side. Wait, no, the two legs? Wait, no, the hypotenuse is 4? Wait, no, let's check again. Wait, the triangle has a right angle, one leg is 3, the hypotenuse is 4? No, wait, maybe the two legs are 3 and \(x\), and hypotenuse 4? Wait, no, Pythagorean theorem: if \(c\) is hypotenuse, then \(a^2 + b^2 = c^2\). Wait, maybe I got the sides wrong. Wait, the right angle is at the top, so the two legs are 3 and the unknown side, and the hypotenuse is 4? Wait, no, 3 and the unknown side are legs, hypotenuse is 4? Wait, no, 3-4-5 triangle? Wait, no, if hypotenuse is 4, then \(3^2 + x^2 = 4^2\)? Wait, \(9 + x^2 = 16\), so \(x^2 = 7\), \(x = \sqrt{7} \approx 2.6\)? Wait, no, maybe I mixed up legs and hypotenuse. Wait, the triangle: the right angle is at the top, so the vertical leg is 3, the hypotenuse is 4? No, that can't be, because hypotenuse must be longer than legs. Wait, maybe the hypotenuse is 4, and one leg is 3, so the other leg is \(\sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7} \approx 2.6\)? Wait, but 3-4-5 is a right triangle, but here hypotenuse is 4, leg 3, so the other leg is \(\sqrt{7}\). Wait, maybe I made a mistake. Wait, let's re-express: let the unknown side be \(x\), right angle between 3 and \(x\), so \(3^2 + x^2 = 4^2\). So \(x^2 = 16 - 9 = 7\), so \(x = \sqrt{7} \approx 2.6\) (rounded to nearest tenth). Wait, but maybe the hypotenuse is the side of length 4, and one leg is 3, so the other leg is \(\sqrt{4^2 - 3^2} = \sqrt{7} \approx 2.6\).

Step2: Apply Pythagorean theorem

\(a = 3\), \(c = 4\) (hypotenuse), so \(b = \sqrt{c^2 - a^2} = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7} \approx 2.6\) (rounded to nearest tenth).

Answer:

\(2.6\)