(D) a parallelogram
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
In ∆ABC,
D and E are midpoints of AB and AC
By midpoint theorem,
DE ∥ BC and DE = 1/2BC
⇒ DE = 1/2[BP + PO + OQ + QC]
∵ P and Q are midpoints of OB and OC
⇒ DE = 1/2[2PO + 2OQ]
⇒ DE = PO + OQ
⇒ DE = PQ …(1)
Now,
In ∆AOC,
Q and C are midpoints of OC and AC
By midpoint theorem,
EQ ∥ AO and EQ = 1/2AO …(2)
Similarly,
In ∆AOB,
D and P are midpoints of AB and OB
By midpoint theorem,
PD ∥ AO and PD = 1/2AO …(3)
Also,
By midpoint theorem,
DE ∥ BC and DE = 1/2BC …(4)
From (2) and (3),
EQ ∥ PD and EQ = PD
From (1) and (4),
DE ∥ BC
DE ∥ PQ and DE = PQ
Hence, DEPQ is a parallelogram
