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Question

In the figure, ABCD is a parallelogram in which P is the mid -point of DC and Q is a point on Ac such that CQ=14AC If PQ produced meets BC at R, prove that R is a mid -point of BC.

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Solution

Given : In || gm ABCD,

P is the mid-point of DC and Q is a point on AC such that CQ=14AC. PQ is produced meets BC at R.

To prove : R is mid point of Bc

Constrction : Join BD

Proof : In || gm ABCD,

Diagonal AC and BD bisect each other at O

AO = OC =12AC ....(i)

In ΔOCD,

P and Q the mid-points of CD and CO

PQ || OD and PQ=12OD

In ΔBCD,

P is mid -point of DC and PQ || OD (Proved above)

Or PR || BD

R is mid -point BC.


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