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