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 ____________.

Answer 1

Given:
D and E are the mid-points of the sides AB and AC of ∆ABC.
O is any point on side BC
P and Q are the mid-points of OB and OC respectively.



Using mid-point theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is equal to the half of it.

In ∆ABC,
DE || BC || PQ ...(1)
DE = $\frac{1}{2}$BC
⇒ DE = $\frac{1}{2}$(BP + PO + OQ + QC)
⇒ DE = $\frac{1}{2}$(2OP + 2OQ)
⇒ DE = OP + OQ
⇒ DE = PQ ...(2)

In ∆AOB,
DP || AO ...(3)
DP = $\frac{1}{2}$AO ...(4)


In ∆AOC,
EQ || AO ...(5)
EQ = $\frac{1}{2}$AO ...(6)

From (3) and (5),
DP || EQ

From (4) and (6),
DP = EQ

Thus, DE || PQ and DP || EQ
DP = EQ and DE = PQ

Hence, DEQP is a parallelogram.