D, E and F are respectively the mid-points of sides AB, BC and CA of Δ ABC. Find theratio of the areas of Δ DEF and Δ ABC

D and E are the mid-points of ΔABC

DE parallel AC

And,

    In ΔBED and ΔBCD

∠BED =∠ BCA     (Corresponding angles)

∠BDE = ∠BAC      (Corresponding angles)

∠EBD = ∠CBA      (Common angles)

Therefore,

ΔBED ~ ΔBCA    (AAA similarity)

Similarly,

And,

Also,

ar(ΔDEF) = ar(ΔABC) – [ar(ΔBED) + ar(ΔCFE) + ar(ΔADF)]

ar(ΔDEF) = ar(ΔABC) – 3/4 ar(ΔBEC)

=1/4ar(ΔBEC)