The complex number `z_(1),z_(2),z_(3)` are the vertices A, B, C of a parallelogram ABCD, then the fourth vertex D is:

Options

A. `(1)/(2)(z_(1)+z_(2))`
B. `(1)/(4)(z_(1)+z_(2)+z_(3)+z_(4))`
C. `(1)/(3)(z_(1)+z_(2)+z_(3))`
D. `z_(1)+z_(3)-z_(2`

Answer

To find the fourth vertex \( D \) of the parallelogram \( ABCD \) given the complex numbers \( z_1, z_2, z_3 \) representing the vertices \( A, B, C \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the vertices**: Let \( z_1 \) be the complex number representing vertex \( A \), \( z_2 \) representing vertex \( B \), and \( z_3 \) representing vertex \( C \).

2. **Use the midpoint property**: In a parallelogram, the midpoints of the diagonals are the same. Therefore, the midpoint of diagonal \( AC \) must equal the midpoint of diagonal \( BD \).

- The midpoint of \( AC \) is given by:
\[ \text{Midpoint of } AC = \frac{z_1 + z_3}{2} \]

- The midpoint of \( BD \) is given by:
\[ \text{Midpoint of } BD = \frac{z_2 + z_4}{2} \]

3. **Set the midpoints equal**: Since the midpoints are equal, we can set the two expressions equal to each other:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]

4. **Eliminate the fractions**: Multiply both sides by 2 to eliminate the fractions:
\[ z_1 + z_3 = z_2 + z_4 \]

5. **Solve for \( z_4 \)**: Rearranging the equation to isolate \( z_4 \) gives:
\[ z_4 = z_1 + z_3 - z_2 \]

Thus, the fourth vertex \( D \) is given by:
\[ z_4 = z_1 + z_3 - z_2 \]

### Conclusion:
The fourth vertex \( D \) of the parallelogram \( ABCD \) is represented by the complex number \( z_4 = z_1 + z_3 - z_2 \).