ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ=1/4AC. If PQ produced meet BC att R, prove that R is the midpoint of BC.
Here is the answer to your query.
Given : ABCD is a parallelogram and P is the mid point of DC.
To prove : R is the mid point of BC.
Constriction : Join B and D and suppose it cut AC at O.
Proof : Now AC (Diagonals of a parallelogram bisect each other) .......(1)
and CD = ...........(2)
From (1) and (2) we get
In ΔDCO, P and Q are mid points of DC and OC respectively.
∴ PQ || DO (mid point theorem)
Also in ΔCOB, Q is the mid point of OC and PQ || AB
∴ R is the mid point of BC (Converse of mid point theorem)