A triangle has corners at #(6 ,7 )#, #(2 ,6 )#, and #(5 ,2 )#. How far is the triangle's centroid from the origin?

Question

A triangle has corners at #(6  ,7  )#, #(2   ,6  )#, and #(5   ,2  )#.  How far is the triangle's centroid from the origin?

Claire Akin · Accepted Answer

Distance of centroid from origin is

#OG = d = sqrt((13/3)^2 + 5^2) = color(purple)(6.6165)#

James Atkins · Answer

undefined The centroid of a triangle is the point where the three medians intersect. To find the centroid of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3), the coordinates of the centroid are ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).

Given the coordinates of the vertices as (6, 7), (2, 6), and (5, 2), the centroid can be found by averaging the x-coordinates and y-coordinates:

x-coordinate of centroid = (6 + 2 + 5)/3 = 13/3
y-coordinate of centroid = (7 + 6 + 2)/3 = 15/3

Thus, the centroid of the triangle is at the point (13/3, 5).

The distance between the origin (0,0) and the centroid can be found using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates of the origin (0,0) and the centroid (13/3, 5), we get:

Distance = √((13/3 - 0)^2 + (5 - 0)^2) = √((13/3)^2 + 5^2) = √(169/9 + 25/1) = √(169/9 + 225/9) = √(394/9)

So, the distance from the origin to the centroid of the triangle is √(394/9).