Question

find the length of the unknown side of the triangle. the length of the unknown side of the triangle is (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Apply the Pythagorean theorem

For a right - triangle with legs \(a = 2\) and \(b = 6\) and hypotenuse \(c\), the Pythagorean theorem is \(c=\sqrt{a^{2}+b^{2}}\).
Substitute \(a = 2\) and \(b = 6\) into the formula: \(c=\sqrt{2^{2}+6^{2}}\).

Step2: Calculate the values inside the square root

First, calculate \(2^{2}=4\) and \(6^{2}=36\). Then \(2^{2}+6^{2}=4 + 36=40\). So \(c=\sqrt{40}\).

Step3: Simplify the square root

We can rewrite \(40\) as \(4\times10\). Then \(\sqrt{40}=\sqrt{4\times10}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 4\), \(b = 10\) and \(a\geq0\), \(b\geq0\)), we get \(\sqrt{4\times10}=\sqrt{4}\cdot\sqrt{10}\). Since \(\sqrt{4}=2\), then \(c = 2\sqrt{10}\).

Answer:

\(2\sqrt{10}\text{ ft}\)