Question

find the length of the unknown side of each triangle. give the exact length and an approximate length.
the exact length of the unknown side is
(simplify your answer. type an exact answer, using radicals as needed. use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Apply Pythagorean theorem

Let the unknown side be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 11\sqrt{5}\) (hypotenuse) and \(a = 9\). Then \(x^{2}+9^{2}=(11\sqrt{5})^{2}\).

Step2: Simplify the equation

First, calculate \(9^{2}=81\) and \((11\sqrt{5})^{2}=11^{2}\times(\sqrt{5})^{2}=121\times5 = 605\). So the equation becomes \(x^{2}+81 = 605\). Then \(x^{2}=605 - 81\).

Step3: Solve for \(x^{2}\) and \(x\)

\(x^{2}=524\). Factor \(524=4\times131\), so \(x=\sqrt{4\times131}=2\sqrt{131}\) (exact value). For the approximate value, \(\sqrt{131}\approx11.45\), then \(x\approx2\times11.45 = 22.9\).

Answer:

The exact length is \(2\sqrt{131}\text{ m}\) and the approximate length is \(22.9\text{ m}\)