To find the centroid of triangle XYZ with vertices X(0, 0), Y(3, 6), and Z(4, 3), we can use the formula for the centroid \((G_x, G_y)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ G_x = \frac{x_1 + x_2 + x_3}{3} \] \[ G_y = \frac{y_1 + y_2 + y_3}{3} \]
Substituting the coordinates of the vertices:
- For vertex X: \((x_1, y_1) = (0, 0)\)
- For vertex Y: \((x_2, y_2) = (3, 6)\)
- For vertex Z: \((x_3, y_3) = (4, 3)\)
Calculating \(G_x\):
\[ G_x = \frac{0 + 3 + 4}{3} = \frac{7}{3} \]
Calculating \(G_y\):
\[ G_y = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]
Thus, the coordinates of the centroid are:
\[ \text{centroid} = \left(\frac{7}{3}, 3\right) \]
Therefore, the centroid of triangle XYZ is at \(\left(\frac{7}{3}, 3\right)\). This point is where the medians meet, thus proving they meet at a single point.