To find the centroid of triangle XYZ with vertices at \( X(0,0) \), \( Y(3,6) \), and \( Z(4,3) \), we use the formula for the centroid \( G \), which is given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices of the triangle.
Substituting the coordinates of the vertices into the formula:
- \( x_1 = 0 \), \( y_1 = 0 \) (Point X)
- \( x_2 = 3 \), \( y_2 = 6 \) (Point Y)
- \( x_3 = 4 \), \( y_3 = 3 \) (Point Z)
We can calculate the x-coordinate of the centroid as follows:
\[ x_G = \frac{0 + 3 + 4}{3} = \frac{7}{3} \]
Next, we calculate the y-coordinate of the centroid:
\[ y_G = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]
Thus, the coordinates of the centroid \( G \) are:
\[ G\left( \frac{7}{3}, 3 \right) \]
So, the centroid is at:
\[ \text{centroid} = \left( \frac{7}{3}, 3 \right) \]
This proves that the medians intersect at a single point, which is the centroid of triangle XYZ.