In the given figure, in triangle ABC, BC = 8 cm, AC = 9 cm and AB = 2 cm. AE = 24 cm, BF = 32 cm and DC = 27 cm. What is the ratio of the area of triangle DEA and area of triangle DLF ?

4 : 9

Let, ∠DAE = θ

then, ∠BAC = 180° - θ

and let ∠FCD = ∝

then, ∠BCA = 180° - ∝

∴ Ratio of area of 

$\frac{ΔDAE}{ΔABC}$ = $\frac{\frac{1}{2}× 18 × 24 × sinθ}{\frac{1}{2} × 12 × 9 × sin (180 - θ)}$

$\frac{ΔDAE}{ΔABC}$ = $\frac{4sinθ}{sinθ}$ = $\frac{4}{1}$

$\frac{ΔDCE}{ΔABC}$ =   = $\frac{9sin∝}{sin∝}$ = $\frac{9}{1}$

Ratio of area,

ΔDAE : ΔABC : ΔDCF

      4 : 1

           1       :     9

      4 : 1       :     9     

∴  Ratio of area $\frac{ΔDAE}{ΔDCF}$ = $\frac{4}{9}$

=  4 : 9