Question

D,E,F are the midpoints of the sides AB,BC and CA respectively of triangle ABC, then
ar(triangle DEF)/ar(triangle ABC) is​

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.

Therefore,we have:

DF=d ×1/2 BC

⇒ DF/BC-1/2............(i)

AC/DE=1/2.........(ii)

AB/EF = 1/2......(iii)

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

Therefore, ΔABC∼ΔEDF [By SSS similarity theorem]

Hence area of ΔABC: area of ΔDEF=4:1