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  4. 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

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Q. $OPQR$ 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

NTA AbhyasNTA Abhyas 2020Straight Lines

Solution:

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 $\left(\frac{a}{2} , \, a\right)$

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