Question

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

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 \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 9\) ft and \(b = 6\) ft, and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 9\) and \(b = 6\) into the formula:

$$
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$$

Step3: Simplify the square root

We can simplify \(\sqrt{117}\) by factoring 117. Since \(117=9\times13\), then \(\sqrt{117}=\sqrt{9\times13}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 9\) and \(b = 13\) and \(a\geq0,b\geq0\)), we get \(\sqrt{9\times13}=\sqrt{9}\times\sqrt{13}=3\sqrt{13}\). And since length cannot be negative, \(c = 3\sqrt{13}\) (we take the positive square root because length is a non - negative quantity).

Answer:

\(3\sqrt{13}\) feet