Question

$ABCD$ is a quadrilateral, $E$ is the point of intersection of the line joining the midpoints of the opposite sides. If $O$ is any point and $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=\overrightarrow{xOE},$ then $x$ is equal to

• A. $3$
• B. $9$
• C. $7$
• D. $4$

Answer 1

Answer: (D)

$4$

Let $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b},\overrightarrow{OC}=\overrightarrow{c}$ and $\overrightarrow{OD}=\overrightarrow{d}$.

Therefore,
$\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}$.
The midpoint $P$ of $AB$, is $\frac{\overrightarrow{a}+\overrightarrow{b}}{2}$.
The position vector of the midpoint $Q$ of $CD$, is $\frac{\overrightarrow{c}+\overrightarrow{d}}{2}$.

Therefore, the position vector of the midpoint of $PQ$ is $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}}{4}$.

Similarly, the position vector of the midpoint of $RS$ is $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}}{4}$, i.e., $\overrightarrow{OE}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}}{4}\Rightarrow x=4$

Hence, option '$D$' is correct.