Question

if D,E,F are the midpoints of the sides AB,BC,CA then find the ratio of the area of ∆DEF and∆ABC

Answer 1

Given in ΔABC, D, E and F are midpoints of sides AB, BC and CA respectively.
BC = EC
Recall that the line joining the midpoints of two sides of a triangle is parallel to third side and half of it.
Hence DF = (1/2) BC
⇒ (DF/BC) = (1/2)  → (1)
Similarly, (DE/AC) = (1/2)  → (2)
(EF/AB) = (1/2)  → (3)
From (1), (2) and (3) we have
DF/BC = DE/AC = EF/AB = 1/2

But if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar
Hence ΔABC ~ ΔEDF [By SSS similarity theorem]
Ar(DEF) /Ar(ABC) = DF^2/BC^2

(DF/BC)^2 = (1/2)^2 = 1/4

Hence area of ΔDEF : area of ΔABC = 1 : 4