Find the centroid of a triangle with the given vertices. X (8,-1) Y (2,7) Z (5, -3).

(1) The centroid is the point where the medians meet. A median is a segment drawn from a vertex to the midpoint of the side opposite that vertex.

(2) Let M be the midpoint of `bar(XY)` , N the midpoint of `bar(YZ)` , and O the midpoint of `bar(XZ)` .

The coordinates of M are `((2+8)/2,(7+(-1))/2)=(5,3)` . Similarly the coordinates of N are (3.5,2) and O (6.5,-2).

(3) The centroid is the point where the lines through (2,7) and (6.5,-2),(8,-1) and (3.5,2), and (5,-3) and (5,3) meet.

-- The equation for `bar(YO)` : the slope is -2, so y=-2x+11. (Using the point-slope formula)
-- The equation for `bar(ZM)` : both x-coordinates are 5, so x=5.
-- The equation for `bar(XN)` : the slope is `-2/3` so `y=-2/3x+(13)/3`

(4) We find the intersection of the lines: since x=5 we use substitution to find the intersection to be (5,1).

Thus the centroid is at (5,1).