Question

find the length of the third side. if necessary, round to the nearest tenth.
right triangle with legs 12 and one leg, hypotenuse 18 (or vice versa, based on the image)

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 hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\) (if \(c\) is the hypotenuse) or \(b^{2}=c^{2}-a^{2}\) (if we want to find a leg, where \(c\) is the hypotenuse and \(a\) is one of the legs). Here, the hypotenuse is \(18\) and one of the legs is \(12\), and we want to find the other leg (let's call it \(x\)).

Step2: Apply the Pythagorean theorem

Using the formula \(x^{2}=c^{2}-a^{2}\), where \(c = 18\) and \(a=12\). So \(x^{2}=18^{2}-12^{2}\).
First, calculate \(18^{2}=324\) and \(12^{2} = 144\). Then \(x^{2}=324 - 144=180\).

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{180}\). Simplify \(\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx6\times2.24 = 13.4\) (rounded to the nearest tenth).

Answer:

The length of the third side is approximately \(13.4\).