Question

$ABCD$ is a quadrilateral. $E$ is the point of intersection of the line joining the middle points of the opposite sides. If $O$ is any point then $\overline{OA}+\overline{OB}+\overline{OC}+\overline{OD}=$

• A. $4\overline{OE}$
• B. $3\overline{OE}$
• C. $2\overline{OE}$
• D. $\overline{OE}$

Answer 1

The correct option is A $4\overline{OE}$

$Concept-$Line joining the mid-point of opposite sides of a quadrilateral bisect each other and also bisect the diagonals of the quadrilateral, i.e.

$\Rightarrow \overrightarrow{EL}=-\overrightarrow{EN}$ and $\overrightarrow{EM}=-\overrightarrow{EK}$ $[\because$ they are equal and opposite$]$...............$A$

And, $\Rightarrow \overrightarrow{EA}=-\overrightarrow{EC}$ and $\overrightarrow{EB}=-\overrightarrow{ED}$ $[\because$ they are equal and opposite$]$................$B$

In $△OEA$,

$\Rightarrow \overrightarrow{OE}+\overrightarrow{EA}=\overrightarrow{OA}$...................$I$

In $△OEB$,

$\Rightarrow \overrightarrow{OE}+\overrightarrow{EB}=\overrightarrow{OB}$...................$II$

In $△OEC$,

$\Rightarrow \overrightarrow{OE}+\overrightarrow{EC}=\overrightarrow{OC}$...................$III$

In $△OED$,

$\Rightarrow \overrightarrow{OE}+\overrightarrow{ED}=\overrightarrow{OD}$...................$IV$

Adding equations $I,II,III,IV$,we get

$\Rightarrow \overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=4\overrightarrow{OE}+\overrightarrow{EA}+\overrightarrow{EB}+\overrightarrow{EC}+\overrightarrow{ED}$.........$V$

But from equation $B$ we know that $\overrightarrow{EA}+\overrightarrow{EB}+\overrightarrow{EC}+\overrightarrow{ED}=0$...........$VI$

From equation $V$ and $VI$, we get

$\Rightarrow \overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=4\overrightarrow{OE}$

Hence answer is option $A$