Question

find the length of the third side. if necessary, write in simplest radical form.
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Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 7\), \(b = 8\) and hypotenuse \(c\), the Pythagorean theorem is \(c^{2}=a^{2}+b^{2}\).
Substitute \(a = 7\) and \(b = 8\) into the formula: \(c^{2}=7^{2}+8^{2}\).
Calculate \(7^{2}=49\) and \(8^{2}=64\), then \(c^{2}=49 + 64\).
So \(c^{2}=113\).

Step2: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{113}\).
Since \(113\) has no perfect - square factors (\(113\) is a prime number), \(\sqrt{113}\) is in simplest radical form.

Answer:

\(\sqrt{113}\)