First slide

Cartesian plane

Question

$OPQR$ is a square and M, N are the middle points of the sides PQ and QR, respectively. Then the ration of the area of the square to that of triangle OMN is

Moderate

A

4:1
B

2:1
C

8:3
D

7:3

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, the area of $ΔOMN$ is
$\frac{1}{2} |\begin{matrix} 0 & 0 & 1 \\ a & a / 2 & 1 \\ a / 2 & a & 1 \end{matrix}| = \frac{3 a^{2}}{8}$
The area of the square is $a^{2} .$ Hence, the required ratio is 8:3

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