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.)
the approximate length of the unknown side is
(round to one decimal place as needed.)

Explanation:

Step1: Apply Pythagorean theorem

Let the unknown side be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 17\sqrt{5}\) and \(a = 7\). So \(x^{2}+7^{2}=(17\sqrt{5})^{2}\).

Step2: Expand the equation

\(x^{2}+49 = 17^{2}\times5\). Since \(17^{2}=289\), then \(x^{2}+49=289\times5 = 1445\).

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

\(x^{2}=1445 - 49\). So \(x^{2}=1396\).

Step4: Find the exact value of \(x\)

\(x=\sqrt{1396}\). Simplify \(\sqrt{1396}=\sqrt{4\times349}=2\sqrt{349}\).

Step5: Find the approximate value of \(x\)

\(\sqrt{349}\approx18.68\), then \(x = 2\sqrt{349}\approx2\times18.68 = 37.4\).

Answer:

The exact length of the unknown side is \(2\sqrt{349}\) m. The approximate length of the unknown side is \(37.4\) m.