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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