Question

derivation of centroid formula

Answer 1

Sol: Let centroid of the triangle the coordinates of whose vertices are given by A(x1, y1), B(x2, y2) and C(x3, y3) respectively. A centroid divides the median in the ratio 2:1. Hence, since ‘G’ is the median so that  AG/AD = 2/1. Since D is the midpoint of BC, coordinates of D are ((x2 + x3)/2, (y2 + y3)/2) By using the section formula, the coordinates of G are ([2(x2+x3)/2) +1× x1] / 2+1, ([2(y2+y3)/2) +1.y1] / 2+1) ∴Coordinates of the centroid G are[ (x1+x2+x3) / 3, (y1+y2+y3 ) / 3].

Answer 2

Let centroid of the triangle the coordinates of whose vertices are given by A(x1, y1), B(x2, y2) and C(x3, y3) respectively.

A centroid divides the median in the ratio 2:1.

Hence, since ‘G’ is the median so that AG/AD = 2/1.

Since D is the midpoint of BC, coordinates of D are ((x2 + x3)/2, (y2 + y3)/2)

By using the section formula, the coordinates of G are

([2(x2+x3)/2) +1× x1] / 2+1, ([2(y2+y3)/2) +1.y1] / 2+1)

∴Coordinates of the centroid G are[ (x1+x2+x3) / 3, (y1+y2+y3 ) / 3].