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 lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if we want to find the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if we want to find a leg), where \(c\) is the hypotenuse (the side opposite the right angle) and \(a\) and \(b\) are the legs. In this triangle, the two legs are of lengths \(a = 4\) and \(b=2\), and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Using the formula \(c=\sqrt{a^{2}+b^{2}}\), substitute \(a = 4\) and \(b = 2\) into the formula.
First, calculate \(a^{2}+b^{2}\):
\(a^{2}=4^{2}=16\) and \(b^{2}=2^{2} = 4\). Then \(a^{2}+b^{2}=16 + 4=20\).
Then, \(c=\sqrt{20}\). We can simplify \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\approx2\times2.24 = 4.5\) (rounded to the nearest tenth).

Answer:

The length of the third side is approximately \(4.5\).