Question

ABCD is a parallelogram in which people is the midpoint of DC and Q is a point on AC such that CQ = 1/4 AC .if PQ produced meets BC at R , PROVE THAT R IS THE MIDPOINT OF BC.

Answer 1

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Answer 2

Question :

ABCD is a parallelogram in which P is the midpoint of DC and Q is point on AC such that CQ = 1/4AC .If PQ produced meets BC at R, prove that R is the midpoint of BC.

Given :

P is the midpoint of DC

CQ = 1/4 AC

To prove :

R is the midpoint of BC

Construction :

Join BD, suppose it meets AC at S .

Proof :

∵ AO = OC = 1/2 AC

Now, CQ = 1/4 AC

               = 1/4(2*OC)

               = OC/2

Q is the midpoint of CO.

In ∆CDO,

∵ P is the midpoint of DC and Q is the midpoint of CO

∴ PR parallel DB and QR parallel OB

Now, in ∆COB,

∵ Q is the midpoint of CO and QR parallel OB

∴ R is the midpoint of BC

[ By converse of Midpoint theorem ]

Hence proved.