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 Pythagorean theorem

Let the unknown side be \(a\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 11\) (hypotenuse) and \(b = 5\). So \(a^{2}+5^{2}=11^{2}\).

Step2: Solve for \(a^{2}\)

\(a^{2}=11^{2}-5^{2}\).
Using the formula \(x^{2}-y^{2}=(x + y)(x - y)\), we have \(a^{2}=(11 + 5)(11 - 5)=16\times6 = 96\).

Step3: Find \(a\)

\(a=\sqrt{96}\).
Simplify \(\sqrt{96}=\sqrt{16\times6}=4\sqrt{6}\).

Answer:

\(4\sqrt{6}\text{ m}\)