O P Q R is a square and M ,   N are the middle points of the sides P Q and Q R respectively, then the…

Taking the coordinates of vertices $O, P, Q, R$ as $(0,0), (a,0), (a,a), (0,a)$, respectively.

$\therefore$ The coordinates of $M$ is $\left(a,\frac{a}{2}\right)$ and $N$ is $\left(\frac{a}{2},a\right)$

$\therefore$ Area of $\mathrm{\Delta }OMN=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ a& a/2& 1\\ a/2& a& 1\end{array}\right|=\frac{3a^2}{8}$

and area of the square $=a^2$

$\therefore$ The required ratio is $8:3$.

Hence, option (c) is correct.