Question

Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar.(AOD) = ar.(BOC). Prove that ABCD is a trapezium.

Answer 1

Data: Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar.(AOD) = ar.(BOC). 

To Prove: ABCD is a trapezium. 

Proof: ar.(∆AOD) = ar.(∆BOC) (Data) 

Adding ar.(∆ODC) on both sides, 

ar.(∆AOD) + ar.(∆ODC) = ar.(∆BOC) + ar.(∆ODC) 

∴ ar.(∆ADC) = ar.(∆BDC) 

Area of these triangles are equal, they are on base DC and in between DC, AB straight lines. 

∴ DC || AB. 

Now, one pair of opposite sides of quadrilateral are parallel, hence ABCD is a trapezium.