Question

$D$ and $E$ are the mid-points of the sides $AB$ and $AC$ of $ABC$ and $O$ is any point on side $BC.O$ is joined to $A$. If $P$ and $Q$ are the mid-points of $OB$ and $OC$ respectively, then $DEQP$ is:

• A. a square
• B. a rectangle
• C. a rhombus
• D. a parallelogram

Answer 1

The correct option is D a parallelogram

Given:

$ABC$ is a triangle, $D$ & $E$ are the respective mid points of $AB$ &$AC$.

$O$ is any point on $BC$, $P$ & $Q$ are mid points of $OB$ & $OC$ respectively.

To find:

Type of quadrilateral of $DEQP$.

Solution:

In $\Delta ABC,D\&E$ are the respective mid points of $AB$ & $AC.$

$\therefore DE\parallel \left(BCorPQ\right)$ ....(1)

Again in $\Delta ABO,P\&D$ are the respective mid points of $AB$ & $BO$

$\therefore DP\parallel AO$ ... (2)

Similarly, in $\Delta ACO,EQ\parallel AO$ ...(3)

$\therefore$ From (2) & (3), we get

$DP\parallel EQ$ ... (4)

$\therefore$ From (1) & (4), we have

$DE\parallel PQ$ and $DP\parallel EQ$

So, $DEQP$ is parallelogram.

Hence, option D.