A triangle has corners at #(1 ,5 )#, #(9 ,4 )#, and #(6 ,7 )#. How far is the triangle's

The centroid of a triangle is the point of intersection of its medians. Centroid divides the medians in the ratio 2 : 1.

If the vertices of a triangle are (x1,y1), (x2,y2), (x3-y3), then the centroid of the triangle is

#G(x,y) = color(red)((x_1 + x_2 + x_3) / 3, (y_1 + y_2 + y_3) / 3)#

#G_x = (1 + 9 + 6) / 3 = 16/3#

#G_y = (5 + 4 + 7) / 3 = 16/3#

Distance of G from origin O is given by the distance formula,

#OG = sqrt((G_x - O_x)^2 + (G_y - O_y)^2) = sqrt(((16/3)-0)^2 + ((16/3-0)^2))#

#OG = sqrt(32/3) = color(blue)(4sqrt(2/3) ~~ 3.266)#