In the given figure `DE||BC and AD:DB =5:4`
Find the ratio `ar (Delta DEF) : ar (Delta CFB)`

Explanation

Since DE||BC
In `triangleADE and triangleABC`
`angle5=angle6` (corresponding angles)
`angleaA = angleA` (common) ,
`triangleADE~triangleABC` (AA corollary)
`(AD)/(AB)=)(DE)/(BC)`
` Rightarrow " " (AD)/(AD+DB) = (DE)/(BC) Rightarrow (BC)/(DE) = (AD+BD)/(AD)`
` Rightarrow" " (BC)/(DE) = 1+ (BD)/(AD) Rightarrow (BC)/(DE) = 1+4/5 " " ( (AD)/(BD) = 5/4`, given )
` (BC)/(DE) = 9/5`
`(DE)/(BC)= 5/9` ...(1)
In ` triangle DEF and triangleCFB`
`angle4 = angle3` ( alternate angles as DE||BC)
and ` angle2= angle1 ` ( vertically opposite angles)
`triangleDEF ~ triangleCBF` (AA corollary)
`(ar(triangleDEF))/(ar(triangleCBF))= (DE^(2))/(BC^(2))= (5/9)^(2)` (area ratio theorem)
Hence , `(ar(triangleDEF))/(ar(triangleCBF))= 25/81`