Question

Question 10
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

In $\Delta ABC$, D and E are the mid-points of sides AB and AC , respectively. By mid-point theorem,

DE ∥ BC

And $DE=\frac{1}{2}BC$
Then, $DE=\frac{1}{2}[BP+PO+OQ+QC]$
$DE=\frac{1}{2}[2PO+2OQ]$
{Since , P and Q are the mid-points of OB and OC respectively]
$\Rightarrow DE=PO+OQ\Rightarrow DE=PQ$
Now, in $\Delta AOC,Q$ and E are the mid-point theorem] ......(iii)

Similarly, in $\Delta AOB,PD\parallel AO$ and $PD=\frac{1}{2}AO$

[ By mid-point theorem] ….(iv)

From eqs. (iii) and (iv)

EQ ∥ PD and EQ = PD

From Eqs .(i) and (ii) DE ∥ BC ( or DE ∥ PQ) and DE = PQ

Hence, DEQP is a parallelogram.