Question

The medians BE and CF of a triangle ABC intersect at G. Prove that the area of △ GBC = area of the quadrilateral AFGE.

Answer 1

Given :

According to the given details

BE & CF are medians

E is the midpoint of AC.

F is the midpoint of AB.

To prove:

area of △ GBC = area of the quadrilateral AFGE.

Proof:

F is the midpoint of AB

∴ ΔBCE = ΔBEA … ( i )

ΔBCF = ΔCAF

Construct:

Join EF,

By midpoint theorem,

We get FE || BC

Δ on the same base and between same parallels are equal in area

∴ ΔFBC = ΔBCE

ΔFBC – ΔGBC = ΔBCE – ΔGBC

⇒ ΔFBG = ΔCGE (ΔGBC is common)

⇒ ΔCGE = ΔFBG …( ii )

Subtracting equation (ii) from (i)

We get,

ΔBCE – ΔCGE = ΔBEA – ΔFBG

∴ ΔBGC = Quadrilateral AFGE.

Hence Proved.