Question

find the length of the third side. if necessary, write in simplest radical form.
8
8
answer attempt 1 out of 2
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Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle with two legs of length 8 (since it has a right - angle symbol and two sides are labeled 8).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 8\) and \(b = 8\). So we substitute these values into the formula:
\(c^{2}=8^{2}+8^{2}\)
\(c^{2}=64 + 64\)
\(c^{2}=128\)

Step3: Simplify the radical

We simplify \(\sqrt{128}\). We can factor 128 as \(64\times2\), and since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{128}=\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\)

Answer:

\(8\sqrt{2}\)