Q. 103.8( 8 Votes )

# D and E are the m

Answer :

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

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