Question

find the length of the third side. if necessary, round to the nearest tenth.
answer attempt 1 out of 2
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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, \(a = 9\) and \(b = 6\), and we need to find the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 9\) and \(b = 6\) into the formula \(c=\sqrt{a^{2}+b^{2}}\). First, calculate \(a^{2}\) and \(b^{2}\): \(a^{2}=9^{2}=81\) and \(b^{2}=6^{2}=36\). Then, \(a^{2}+b^{2}=81 + 36=117\). So, \(c=\sqrt{117}\).

Step3: Simplify and round

We know that \(\sqrt{117}\approx10.8\) (because \(10.8^{2}=116.64\) and \(10.9^{2}=118.81\), and \(117\) is closer to \(116.64\) when considering rounding to the nearest tenth).

Answer:

\(10.8\)