Question

what is the value of ( n ) in the triangle below?

( 8 sqrt { 2 } )
( 4 sqrt { 2 } )
( \frac { sqrt { 2 } } { 8 } )
16

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 8\), \(b = 8\) and hypotenuse \(n\), the Pythagorean theorem is \(a^{2}+b^{2}=n^{2}\).
Substitute \(a = 8\) and \(b = 8\) into the formula: \(n^{2}=8^{2}+8^{2}\).

Step2: Calculate \(n^{2}\)

\(n^{2}=64 + 64\).
\(n^{2}=128\).

Step3: Solve for \(n\)

\(n=\sqrt{128}\).
Simplify \(\sqrt{128}=\sqrt{64\times2}\).
Since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 64\), \(b = 2\)), then \(n=\sqrt{64}\times\sqrt{2}\).
\(n = 8\sqrt{2}\).

Answer:

\(8\sqrt{2}\)