Question

D,E&F are the midpoints of the sides BC,CA&AB respectively,of triangle ABC. Determine the ratio of the areas of triangle DEF and ABC

Answer 1

Given in ΔABC, D, E and F are midpoints of sides AB, BC and CA respectively.BC = ECRecall 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 similarHence ΔABC ~ ΔEDF [By SSS theorem]ar(triangle DEF)/ar(triangle ABC)
=DF^2/BC^2
=(DF/BC)^2
=(1/2)^2
=1/4Hence area of ΔDEF : area of ΔABC = 1 : 4
I hope this is correct.

Answer 2

here is your answer by Sujeet,



that's all