Question

if D,E,F are midpoints of sides AC,BCand AB respectively of ∆ ABC then area of triangle DEF: ar ∆ ABC is​

Answer 1

Step-by-step explanation:

given,

In triangle ABC

D,E,F are midpoints of sides BC,CA,AB respectively

from triangle ABC

AF =FB ×1

AF/FB=1×1 (Equation 1 )

AE=EC

AE/EC=1/1 (Equation2)

from 1 and 2

AF/FB=AE/EC

I.e. the FE divides AB and AC in the same ratio

therefore FE II BC (Equation3)

similarly

DE II AB (Equation4)

and

FD II AC (Equation5 )

from 3 4 5

AFDE is a parallelogram

in parallelogram opposite angles are equal

so ,

angle A = angle FDE (Equation6)

BDEF is a parallelogram

in parallelogram opposite angles are equal

so,

angle B = angle FED(Equation7)

from 6 and 7

in triangle ABC and DEF

Angle A = Angle FDE

Angle B= Angle FED

According to AAA Similarity

Triangle DEF similar to triangle ABC

We know that the ratios of areas of two similar triangles are equal to ratio of squares of their corresponding sides

triangle DEF/triangle ABC =7E square/BCsquare

= 1/2 BC square/ BC square (in a triangle the line joining mid point of two sides is parallel to third side and half of it)

= (1/2)square × (BC/BC)square

TRIANGLE DEF/TRIANGLE ABC =1/4

THEREFORE TRIANGLE DEF :TRIANGLE ABC=1:4