Question

$OPQR$ is a square, $M$ and $N$ are the middle points of the sides $PQ$ and $QR$ respectively.

• A. ratio of the areas of quadrilateral $OMNR$ and the square is $5:8$
• B. ratio of the areas of the $\Delta MQN$ and $\Delta OMN$ is $1:3$
• C. ratio of the areas of the $\Delta OPM$ and $\Delta ORN$ is $1:1$
• D. ratio of the areas of quadrilateral $OMQN$ and the square is $1:2$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A ratio of the areas of quadrilateral $OMNR$ and the square is $5:8$
B ratio of the areas of the $\Delta MQN$ and $\Delta OMN$ is $1:3$
C ratio of the areas of the $\Delta OPM$ and $\Delta ORN$ is $1:1$
D ratio of the areas of quadrilateral $OMQN$ and the square is $1:2$

Let $O\equiv (0,0),P\equiv (a,0),Q\equiv (a,a),R\equiv (0,a)$ be the vertices of the square $OPQR$ of area $=a^2$
And $M\equiv (a,\frac{a}{2}),N\equiv (\frac{a}{2},a)$
Area of $△OMN=\frac{3a^2}{8}$
Area of $△MQN=\frac{a^2}{8}$
Area of $△OPM=$ Area of $△ORN=\frac{a^2}{4}$
Area of quadrilateral $OMQN=a^2-\frac{a^2}{2}=\frac{a^2}{2}$
Area of quadrilateral $OMNR=\frac{3a^2}{8}+\frac{a^2}{4}=\frac{5a^2}{8}$