To find the coordinates of the third vertex of a triangle when the coordinates of the centroid and two vertices are known, we can use the properties of the centroid. The centroid of a triangle is the point where the medians intersect, and it divides each median into a ratio of 2:1. The coordinates of the centroid ( G ) of a triangle with vertices ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ) are given by the formula:

[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]

Given that the coordinates of the centroid ( G ) are ( (3, 2) ) and the coordinates of two vertices ( A ) and ( B ) are ( (1, 1) ) and ( (2, 5) ) respectively, we can set up the following equations:

[ 3 = \frac{1 + 2 + x_3}{3} ] [ 2 = \frac{1 + 5 + y_3}{3} ]

First, solve for ( x_3 ):

[ 3 = \frac{1 + 2 + x_3}{3} ] [ 9 = 1 + 2 + x_3 ] [ 9 = 3 + x_3 ] [ x_3 = 6 ]

Next, solve for ( y_3 ):

[ 2 = \frac{1 + 5 + y_3}{3} ] [ 6 = 1 + 5 + y_3 ] [ 6 = 6 + y_3 ] [ y_3 = 0 ]

Thus, the coordinates of the third vertex ( C ) are ( (6, 0) ). This solution demonstrates how the centroid formula can be used to find the missing vertex of a triangle when the coordinates of the other two vertices and the centroid are known. The method involves setting up and solving linear equations based on the given coordinates and the properties of the centroid.