Question

if D,E and F are mid points of sides BC ,CA and AB of a triangle ABC respectively then the ratio of the areas of triangle DEF and ABC is​

Answer 1

Step-by-step explanation:

given

D,E and F are the midpoint of the sides AB,BC and CA of the ∆ ABC.

Join the point D,E,F.

In ∆ ABC we have

By the mid point theorem we have

DF = 1 /2 BC

DE = 1/2 CA.............. 1 Eq.

EF = 1/2 AB

IN ∆ DEF and ∆ CAB

DF / BC= DE /CA=EF/AB= 1/2

FROM EQUATION 1

angle ∆ DEF ~ ∆ CAB

by SSS similarity criterion

ar ∆ DEF / ar ∆ CAB = DE² / CA²

from equation 1

ar ∆ DEF / ar ∆ CAB = 1/4

ar ∆ DEF / ar ∆ ABC = 1/4

ar ∆ CAB / ar ∆ ABC = 1/4

ar ∆ DEF / ar ∆ ABC = 1:4

Hence the ratio of the area ∆ DEF area ∆ ABC is 1 : 4