Question

A triangle is divided into seven triangles. The areas of four of them are 420 `cm^2`, 80 `cm^2`, 60 `cm^2` and 30 `cm^2` as shown in the diagram on the right. Find the area of triangle AEF, in `cm^2`?

Answer 1

Area `AEF=1512 cm^2`

Explanation:

When a line from one of the vertex of a triangle, meets the opposite side, the ratio between the areas of the two triangles is same as the ratio of the two parts in which it divides the opposite side.

By corollary, in a quadrilateral whose diagonals divide it in four triangles, areas of two adjacent triangles are in the same ratio as the ratio of other two triangles.

Let `DeltaXYZ` denotes area of triangle `XYZ`.
Given `DeltaBDG : DeltaGDE = 30 : 60 = 1:2`
`=> DeltaBDC : DeltaDCE = DeltaBDC:80=1:2`
`=>DeltaBDC=40`

As `DeltaGCB:BCF=70:420=1:6`
`=> GB:BF=1:6`
`=> DeltaGAB:DeltaBAF=1:6`
let `DeltaGAB=1a, and DeltaBAF=6a`

As `DeltaEBC:DeltaCBF=120:420=2:7`
`=> EC:CF=2:7`
`=> DeltaEAC:DeltaCAF=2:7`
`=> (1a+210):(6a+420)=2:7`
`=> (1a+210)xx7=(6a+420)xx2`
`=> 1470-840=12a-7a`
`=> 5a=630`
`=> a=126`

Area of triangle `AEF=1a+6a+210+420=7a+210+420`
`=7xx126+210+420=1512 cm^2`