Pythagoras’ Theorem

Pythagoras' Theorem (for a right angled triangle) can be written as:  \[ \mbox{(perpendicular height)}^2 + \mbox{(base)}^2 = \mbox{(hypotenuse)}^2 \]

Consider now the right angle triangle with an angle of \(\theta\): 

 

Pythagoras' Theorem now can be written as: 
\begin{eqnarray*}
\mbox{(opposite)}^2 + \mbox{(adjacent)}^2 &=& \mbox{(hypotenuse)}^2\\
a^{2} + b^{2} &=& c^{2} 
\end{eqnarray*}

For example, find the missing side of a triangle with hypotenuse of \(10\) and a base of \(6\).

\begin{eqnarray*}
a^{2}+ b^{2} &=& c^{2} \\
a^{2}+6^{2}&=&10^{2} \\
a^{2}&=& 100 - 36 \\
a^{2}&=& 64 \\
a&=& \sqrt{64} \\
&=& 8
\end{eqnarray*}