1.

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

Answer»

Data: In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. 

To Prove : Line segments AF and EC trisect the diagonal BD. 

Proof: ABCD is a parallelogram. 

AB || DC and AB = DC 

\(\frac{1}{2}\)AB = \(\frac{1}{2}\)DC 

AE = CF 

and AE || CF (∵ AB || CD) 

In AECF quadrilateral, 

∴ AE || CF and AE = CF. 

∴ AECF is a parallelogram. 

∴ AF || EC 

In ∆DQC, 

PF || QC (∵ AF || EC) 

∴ P is the mid-point of DQ. 

∴ DP = PQ ………….. (i) 

In ∆APB, EQ || AP. 

But E is the mid-point of AB (Data) 

∴ Q is the mid-point of PB. 

∴ PQ = QB …………… (ii) 

From (i) and (ii), DP = PQ = QB 

∴ AF and EC line segments trisects the diagonal BD.


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