question_answer

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 \[\Delta OMN\] is

A) 4 : 1     

B)  2 : 1    

C)  8 : 3    

D)  7 : 3

Correct Answer: C

Solution :

Let the coordinates of vertices O, P, Q, R be (0, 0), (a, 0), (a, a), (0, a), respectively. Then, we get the coordinates of M as (a, a/ 2) and those of N as (a / 2, a) Therefore, area of \[\Delta OMN\] is \[\frac{1}{2}\left| \begin{matrix}    0 & 0 & 1  \\    a & a/2 & 1  \\    a/2 & a & 1  \\ \end{matrix} \right|=\frac{3{{a}^{2}}}{8}\] Area of the square is \[{{a}^{2}}.\]Hence, the required ratio is 8 : 3.