ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ=^{1}/_{4}AC. If PQ produced meet BC att R, prove that R is the midpoint of BC.

Dear Student!

Here is the answer to your query.

**Given :** ABCD is a parallelogram and P is the mid point of DC.

Also,

**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)

Cheers!

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