Answer
The ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{\sqrt{2}}{2}$.
Work Step by Step
Consider the provided values, $a=1$ and $b=1$
Substitute $a=1$ and $b=1$ in the equation ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$.
$\begin{align}
& {{c}^{2}}={{1}^{2}}+{{1}^{2}} \\
& {{c}^{2}}=1+1 \\
& {{c}^{2}}=2
\end{align}$
Use the square root property.
$c=\pm \sqrt{2}$
Sides of a triangle are always positive $c>0$. So, the value of $c$ is $\sqrt{2}$.
The ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{a}{c}$.
Substitute $a=1$ and $c=\sqrt{2}$ in the expression $\frac{a}{c}$.
$\frac{a}{c}=\frac{1}{\sqrt{2}}$
Multiply the numerator and denominator by $\sqrt{2}$.
$\begin{align}
& \frac{a}{c}=\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\
& =\frac{\sqrt{2}}{2}
\end{align}$
Therefore, the ratio of the length of the side opposite $\theta $ to the length of the hypotenuse is $\frac{\sqrt{2}}{2}$.
