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: Use position vectors.
With midpoints, $\vec m = \tfrac{\vec p + \vec q}{2}$ and $\vec n = \tfrac{\vec r + \vec s}{2}$.

Step 2: Build MN.
$$2\overline{MN} = \vec r + \vec s - \vec p - \vec q$$

Step 3: Regroup into the sides.
$(\vec s - \vec p) = \overline{PS}$ and $(\vec r - \vec q) = \overline{QR}$, so $2\overline{MN} = \overline{PS} + \overline{QR}$. Comparing, $t = 2$.
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