Answer
The ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{5}{13}$.
Work Step by Step
Consider the provided values, $a=5$ and $b=12$
Substitute $a=5$ and $b=12$ in the equation ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$.
$\begin{align}
& {{c}^{2}}={{5}^{2}}+{{12}^{2}} \\
& {{c}^{2}}=25+144 \\
& {{c}^{2}}=169
\end{align}$
Use the square root property.
$\begin{align}
& c=\pm \sqrt{169} \\
& =\pm 13
\end{align}$
Sides of a triangle are always positive $c>0$. So, the value of $c$ is $13$.
The ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{a}{c}$.
Substitute $a=5$ and $c=13$ in the expression $\frac{a}{c}$.
$\frac{a}{c}=\frac{5}{13}$
Therefore, the ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{5}{13}$.
