Question

If the coordinates of the midpoints of the sides of a triangle are (1,1) ,(2,-3) , (3,4) then find the centroid.

Answer 1

The Centroid is the Point dividing a median in the ratio 2:1.
Let The Triangle have vertices A(1,1) , B(2,-3) , C(3,4).
A Median divides a side in the ratio 1:1.
Let the Median divide side AC into 1:1.
Applying Section Formula,
(x,y)=(m1x2+m2x1/m1+m2 , m1y2+m2y1/m1+m2) = x=3+1/2 =4/2=2 and y=4+1/2=5/2.
Let BC Be the Median. The Centroid Divides BC in the ratio 2:1.
Again, Applying Section Formula on BC:
(x,y)=(m1x2+m2x1/m1+m2 , m1y2+m2y1/m1+m2) = x=2*2+2*1/3=4+2/3=6/3=2 and y=2*5/2+-3*1/3=5-3/3=2/3.
Therefore the Centroid is on the point (2,2/3)