Question

If in a triangle ABC, DEF are the mid point s of sides BC,CA,AB respectively then write the ratio of ar(triangle DEF) : ar(triangle 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 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