Answer to Question #97211 in Analytic Geometry for Aravind

Question #97211
What are the coordinates of the centroid of the triangle with vertices (X1,Y1), (X2,Y2), (X3 ,Y3)
1
Expert's answer
2019-10-24T09:29:25-0400

Let ABC be the triangle where A(x1, y1), B(x2, y2) and C(x3, y3):





Let G be the centroid of triangle ABC.

Let AD be the median of triangle ABC. So, D is the midpoint of BC.

Midpoint of B(x2, y2) and C(x3, y3) is

"D= (\\frac {x_2 +x_3} {2},\\frac {y_2 +y_3} {2})"

We know that centroid divides median in the ratio 2:1. So, centroid G divides the median AD in the ratio 2:1.

We know that coordinate of point P(x, y) that divides the line segment joining  internally in the ratio m:n is

"P(x,y)= (\\frac {mx_2 +nx_1} {m+n},\\frac {my_2 +ny_1} {m+n})"

Here,

"x_1=x_1" ,

"y_1=y_1" ,

"x_2=\\frac{x_2+x_3}{2}" ,

"y_2=\\frac{y_2+y_3}{2}" ,

"m=2" ,

"n=1"

Coordinate of G are

"G(x,y)= \\frac {2 (\\frac {x_2+x_3}{2}) +1(x_1)}{2+1},\\frac {2 (\\frac {y_2+y_3}{2}) +1(y_1)}{2+1} \n=(\\frac {x_2+x_3+x_1}{3},\\frac {y_2+y_3+y_1}{3})"

Hence Coordinate of centroid are

"G(x,y)=(\\frac {x_1+x_2+x_3}{3},\\frac {y_1+y_2+y_3}{3})"


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