`z_1, z_2, z_3,z_4` are distinct complex numbers representing the vertices of a quadrilateral `A B C D` taken in order. If `z_1-z_4=z_2-z_3 and "arg"[(z_4-z_1)//(z_2-z_1)]=pi//2` , the quadrilateral is

Options

A. rectangle
B. rhombus
C. square
D. trapezium

Explanation

To solve the problem, we need to analyze the conditions given for the complex numbers \( z_1, z_2, z_3, z_4 \) representing the vertices of a quadrilateral \( ABCD \).

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
We have two conditions:
- \( z_1 - z_4 = z_2 - z_3 \)
- \( \text{arg}\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2} \)

2. **Using the First Condition**:
From the first condition \( z_1 - z_4 = z_2 - z_3 \), we can rearrange it to:
\[ z_1 + z_3 = z_2 + z_4 \]
This implies that the midpoints of the diagonals \( AC \) and \( BD \) are the same.

3. **Finding the Midpoint**:
The midpoint of diagonal \( AC \) is:
\[ \frac{z_1 + z_3}{2} \]
And the midpoint of diagonal \( BD \) is:
\[ \frac{z_2 + z_4}{2} \]
Since \( z_1 + z_3 = z_2 + z_4 \), we conclude that:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]
This means that the diagonals bisect each other.

4. **Using the Second Condition**:
The second condition \( \text{arg}\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2} \) indicates that the vector \( z_4 - z_1 \) is perpendicular to the vector \( z_2 - z_1 \). This means that the angle between the sides \( AD \) and \( AB \) is \( 90^\circ \).

5. **Conclusion**:
Since the diagonals bisect each other and the angle between two adjacent sides is \( 90^\circ \), we conclude that the quadrilateral \( ABCD \) is a rectangle.

### Final Answer:
The quadrilateral is a **rectangle**.