Question

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

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

Answer 1

Answer: (C)

$8:3$

Let the coordinates of vertices $O,P,Q,R$ be $(0,0),(a,0),(a,a),(0,a)$, respectively.
Area of the square is $a^2$
Coordinates of $M$ are $(a,\frac{a}{2})$ and those of $N$ are $(\frac{a}{2},a)$.
Therefore, area of $\Delta$OMN
$=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ a& \frac{a}{2}& 1\\ \frac{a}{2}& a& 1\end{array}\right|$
$=\frac{3a^2}{8}$
Hence, the required ratio is $8:3$.