Question

$ABCD$ is a quadrilateral and $E$ the point of intersection of the lines joining the middle points of opposite sides. If $O$ is any point, then the resultant of $OA,OB,OC$ and $OD$ is equal to

• A. $2OE$
• B. $OE$
• C. $4OE$
• D. none of these

Answer 1

The correct option is C $4OE$

Let $P,Q,R,S$ be the mid-points of sides $BC,CD,DA,AB$ respectively of a quadrilateral $ABCD$.

BY geometry the figure formed by joining the mid-point $P,Q,R,S$ will be a parallelogram.

Hence, its diagonal will bisect each other, say at $E$.

Now $\because P$ is the mid point of $BC$

$\therefore OB+OC=2OP$ ....(1)

And $R$ is the mid point of $AD$

$\therefore OA+OD=2OR$ ....(2)

Adding (1) and (2), we have

$OA+OB+OC+OD=2(OP+OR)=2.2OE=4OE$

$(\because E$ is mid point of $PR$,

$\therefore OP+OR=2OE)$.