Question

ABCD is a parallelogram . the sides DA and DC are produced upto the points P and Q in such a way that AP = DA and CQ = DC .Prove that, the points P , B , Q are collinear​

Answer 1





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Question



A parallelogram ABCD has P the mid- point of DC and Q intersects AC such that CQ=41AC. PQ produced meets BC at R prove that :

(a) R is the mid-point of BC

(b) PR=21DB

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Solution



Verified by Toppr

Given ABCD is a parallelogram and P is midpoint of DC 

also, CQ=41AC

To Prove : R is mid point of BC

Proof : Now

OC=21AC (Diagonals of parallelogram bisect each other)     ...(i)

and CD=41AC       ...(ii)

From (i) and (ii)

CD=21OC

In ΔDCO P and Q are midpoint of DC and OC Respectively

∴PQ∥DO

Also in ΔCOB Q is midpoint of OC and PQ∥DB

∴R is midpoint of BC

∴ in ΔABCPR∥DB

CDCP=CBCR=BDPR

DBPR=21

∴PR=21DB