Question

find the length of the third side. if necessary, write in simplest radical form.

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (has a right - angle symbol). For a right - triangle, we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). Here, we need to check if the given sides are legs or if one is the hypotenuse. The hypotenuse is the longest side in a right - triangle. The given sides are \(2\) and \(4\). Since \(4>2\), we assume that \(4\) is the hypotenuse (let \(c = 4\)) and one leg \(a = 2\), and we need to find the other leg \(b\).

Step2: Apply the Pythagorean theorem

From \(a^{2}+b^{2}=c^{2}\), we can re - arrange it to solve for \(b\): \(b^{2}=c^{2}-a^{2}\). Substitute \(a = 2\) and \(c = 4\) into the formula: \(b^{2}=4^{2}-2^{2}\). Calculate \(4^{2}=16\) and \(2^{2}=4\). Then \(b^{2}=16 - 4=12\).

Step3: Simplify the radical

To find \(b\), we take the square root of both sides: \(b=\sqrt{12}\). Simplify \(\sqrt{12}\) by factoring \(12 = 4\times3\), so \(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}\).

Answer:

The length of the third side is \(2\sqrt{3}\)