Diagonals of a trapezium PQRS intersect each other at the point 0 ,PQ`abs()`RS and PQ=3 RS. Find the ratio of the areas of `DeltaPOQ` and `DeltaROS`.

Explanation

Given PQRS is a trapezium in which PQ`abs()`Rsand PQ=3 RS
`rArr(PQ)/(RS)=3/1`…(i)

In `DeltaPOQ` and `DeltaROS` `angleSOR=angleQOP` [vertically opposite angles]
`angleSRP=angleRPQ` [alternate angles]
`therefore DeltaPOQ~DeltaROS` [by AAA similarity criterion]
By property of area of similar triangle,
`(ar(DeltaPOQ))/(ar(DeltaSOR))=((PQ)^(2))/((RS)^(2))=((PQ)/(RS))^(2)=(3/1)^(2)` [from Eq.(i)]
`rArr (ar(DeltaPOQ))/(ar(DeltaSOR))=9/1`
Hence, the required ratio is 9 : 1