If one diagonal of the trapezium divides in the ratio 1:2

prove that one of the parallel side is double the other

Given : that diagonal BD divides diagonal AC in AO : OC with 3 : 1.

To prove : AB = 2 CD.

Proof : In AOC and DOC

 ∠AOB =    ∠COD

 ∠OBA =   ∠ODC ( because DC is parallel to AB and DB is the transversal, so these are alternate   ∠s )

 Therefore,  AOB   COD    [ By AA Similarity ]

Now, AO / OC = AB / DC                                ( because in similar triangles sides are propotional )

2 / 1  =  AB / DC                                               ( given that AO : OC = 2 : 1 )

AB = 2 DC