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.
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.