To find the area of a triangle given the coordinates of its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), we can use the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Let's assign the coordinates of the triangle corners:
- \(A (x_1, y_1) = (0, 3)\)
- \(B (x_2, y_2) = (10, 6)\)
- \(C (x_3, y_3) = (6, 3)\)
Substituting these coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 0(6 - 3) + 10(3 - 3) + 6(3 - 6) \right| \]
Let's calculate each term inside the absolute value:
- \(0(6 - 3) = 0\)
- \(10(3 - 3) = 0\)
- \(6(3 - 6) = 6 \times (-3) = -18\)
Now substituting these back into the formula:
\[ \text{Area} = \frac{1}{2} \left| 0 + 0 - 18 \right| = \frac{1}{2} \left| -18 \right| = \frac{1}{2} \times 18 = 9 \]
Thus, the area of the triangle is:
\[ \boxed{9} \text{ square units} \]