Triangles, Quadrilaterals

Question

In triangle ABC, D and E are the point on AC and AB respectively. BD and CE intersect at F. If the areas of triangles EBF, BFC, FDC are 10, 20, 16 respectively. Then area of quadrilateral AEFD is?

Answer 1

let AF meet triangle at X.Now in triangle ABX we have AE/EB + AD/DC = AF/FX...(*)

now, AE/EB = ar(ACE)/ar(ECB) = (ar(ADFE) + ar(DCF))/(ar(EFB) + ar(BFC))

= (ar(ADFE)+16)/(10+20) = (ar(ADFE)+16)/30 ...(1)

&, AD/DC =(ar(ADFE)+ar(EFB))/(ar(BFC)+ar(DFC))

=(ar(ADFE)+10)/(20+16) = (ar(ADFE)+10)/(36) ...(2)

& AF/FX =(ar(ABF)+ar(ACF))/(ar(BFX)+ar(XFC)))

=(ar(EFB)+ar(ADFE)+ar(DFC))/(ar(BFC)

=(10+ar(ADFE)+16)/20 = (ar(ADFE)+26)/20 ...(3)

putting values from(1),(2)&(3)in(*) we get

ar(ADFE)=36.

(here ar(...) represents area of figure in () )