Step-by-step Solution:

To find the area of a triangle given its three vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \), we use the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Given vertices: \[ (x_1, y_1) = (1,2), \quad (x_2, y_2) = (2,5), \quad (x_3, y_3) = (4,3) \] Substituting the values: \[ \text{Area} = \frac{1}{2} \left| 1(5 - 3) + 2(3 - 2) + 4(2 - 5) \right| \] \[ = \frac{1}{2} \left| 1(2) + 2(1) + 4(-3) \right| \] \[ = \frac{1}{2} \left| 2 + 2 - 12 \right| \] \[ = \frac{1}{2} \left| -8 \right| \] \[ = \frac{1}{2} \times 8 = 4 \] Final Answer: \[ \boxed{4} \text{ square units} \]