Question

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = `1/3` (area of ΔABC)

Answer 1

W.K.T the median of a triangle divides it into two triangles of equal area.

In ΔABC, AD is the median

Area(ΔABD) = Area (ΔACD)  ........(1)

In ΔGBC, GD is die median

Area(ΔGBD) = Area (ΔGCD)   ........(2)

Sub (2) from (1) we get

Area(ΔABD) – Area (ΔGBD)

= Area(ΔACD) – Area (ΔGCD)

Area(ΔAGB) = Area(ΔAGC)   ........(3)

Similarly

Area(ΔAGB) = Area(ΔBGC)   ........(4)

From (3) and (4) we get

Area(ΔAGB) = Area(ΔAGC) = Area(ΔBGC)   ........(5)

Now

Area(ΔAGB) + Area(ΔAGC) + Area(ΔBGC) = Area(ΔABC)

⇒ Area(ΔAGB) + Area(ΔAGB) + Area(ΔAGB)

= Area(ΔABC)  .......(Using 5)

⇒ 3Area(ΔAGB) = Area(ΔABC)

⇒ Area(ΔAGB) = `1/3` area(ΔABC) ........(6)

From (5) and (6) we get

Area(ΔAGB) = Area(ΔAGB) = Area(ΔBGC)

= `1/3` area(ΔABC)