Question

In a quadrilateral PQRS, M and N are mid-points of the sides PQ and RS respectively. If $\overline{PS} + \overline{QR} = t\overline{MN}$, then $t =$

• A. $\frac{1}{2}$
• B. $4$
• C. $\frac{3}{2}$
• D. $2$

Hint:

This is a standard vector geometry property: In any quadrilateral, the vector sum of two opposite sides is always equal to exactly twice the vector connecting the midpoints of the other two sides.

Answer 1

The Correct Option is D

Step 1: Name the position vectors.
Let $P,Q,R,S$ have position vectors $\vec{p},\vec{q},\vec{r},\vec{s}$. $M$ is the midpoint of $PQ$ and $N$ of $RS$.
Step 2: Write the midpoints.
$\vec{m}=\dfrac{\vec{p}+\vec{q}}{2}$ and $\vec{n}=\dfrac{\vec{r}+\vec{s}}{2}$.
Step 3: Form vector $MN$.
$\overline{MN}=\vec{n}-\vec{m}=\dfrac{\vec{r}+\vec{s}-\vec{p}-\vec{q}}{2}$.
Step 4: Clear the half.
$2\,\overline{MN}=(\vec{r}+\vec{s}-\vec{p}-\vec{q})$.
Step 5: Regroup into the side vectors.
Group as $(\vec{s}-\vec{p})+(\vec{r}-\vec{q})=\overline{PS}+\overline{QR}$. So $2\,\overline{MN}=\overline{PS}+\overline{QR}$.
Step 6: Compare with the given.
Since $\overline{PS}+\overline{QR}=t\,\overline{MN}$, we read off $t=2$, which is option (4). \[ \boxed{t=2} \]