Question

In triangleABC, D and E are the midpoints of AB and AC respectively. F is any point on BC. Prove that ar(ADFE) = 1/2 ar(ABC)

Answer 1

Given: AD = BD & AE= EC

In Triangle ADE & Triangle FEC,

EC=AE

DE=FC

AD=EF

Therefore,

Triangle ADE and Triangle FEC are congruent to each other. ........⬅eq.1

Therefore,

Area of all the 4 triangle is equal ( from eq.1)...........⬅eq.2

Ar.ABC = Ar. ADE+BDF+FEC+FDE

Ar. ABC = 2 ( Ar. ADE + Ar. FDE)

1/2 Ar. ABC = Ar. ADEF

HENCE PROVED!