Question

$OPQR$ is a square and $M,N$ are the mid points of the sides $PQ$ and $QR$ respectively, then the ratio of the areas of the square and the triangle $OMN$ is

• A. $4:1$
• B. $2:1$
• C. $8:3$
• D. $7:3$

Answer 1

Answer: (C)

Let the coordinates of the vertices $O,P,Q,R$ be $\left(0,0\right),\left(a,0\right),\left(a,a\right),\left(0,a\right),$ respectively. Then, we get the coordinates of $M$ as $\left(a,\frac{a}{2}\right)$ and those of $N$ as $\left(\frac{a}{2},a\right)$


Therefore, area of $\Delta OMN$ is
$\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ a& \frac{a}{2}& 1\\ \frac{a}{2}& a& 1\end{array}\right|=\frac{3a^2}{8}$
Area of the square is $a^2$ .
Hence, the required ratio is $8:3$