Question

If D, E and F are respectively the midpoints of sides AB, BC and CA of $\Delta ABC$ then what is the ratio of the areas of $\Delta DEFand\Delta ABC$?

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 = BC
⇒ → (1)
Similarly, = → (2)
= → (3)
From (1), (2) and (3) we have

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]

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