## Precalculus (6th Edition) Blitzer

The ratio of the length of the side opposite $\theta$ to the length of the hypotenuse is $\frac{5}{13}$.
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 $\frac{5}{13}$.