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 \( G \) of a triangle, which is given by the coordinates:
\[ G\left( x, y \right) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices \( X \), \( Y \), and \( Z \).
Substituting the coordinates of points \( X \), \( Y \), and \( Z \):
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (3, 6) \)
- \( (x_3, y_3) = (4, 3) \)
We calculate the \( x \)-coordinate of the centroid:
\[ x = \frac{0 + 3 + 4}{3} = \frac{7}{3} \]
Next, we calculate the \( y \)-coordinate of the centroid:
\[ y = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]
Therefore, the coordinates of the centroid \( G \) are:
\[ G\left( \frac{7}{3}, 3 \right) \]
So the centroid of triangle \( XYZ \) is:
\[ \text{centroid} = \left( \frac{7}{3}, 3 \right) \]