If `P(z_(1)),Q(z_(2)),R(z_(3)) " and " S(z_(4))` are four complex numbers representing the vertices of a rhombus taken in order on the complex plane, which one of the following is held good?

Options

A. `(z_(1)-z_(4))/(z_(2)-z_(3))` is purely real
B. `(z_(1)-z_(3))/(z_(2)-z_(4))` is purely imaginary
C. `abs(z_(1)-z_(3)) ne abs(z_(2)-z_(4))`
D. `amp((z_(1)-z_(4))/(z_(2)-z_(4))) ne amp((z_(2)-z_(4))/(z_(3)-z_(4)))`

Answer

To solve the problem, we need to analyze the properties of a rhombus in the complex plane. The vertices of the rhombus are represented by the complex numbers \( P(z_1), Q(z_2), R(z_3), S(z_4) \).

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
- In a rhombus, opposite sides are equal and the diagonals bisect each other at right angles.
- The diagonals of a rhombus are not equal in length.

2. **Analyzing the Options**:
- We will check each option one by one.

3. **Option 1**: \( \frac{z_1 - z_4}{z_2 - z_3} \) is purely real.
- Here, \( z_1 - z_4 \) represents one diagonal (let's say \( PS \)) and \( z_2 - z_3 \) represents the other diagonal (let's say \( QR \)).
- Since \( PS \) is parallel to \( QR \), the argument of \( \frac{z_1 - z_4}{z_2 - z_3} \) is 0, which means it is purely real.
- **Conclusion**: This option is correct.

4. **Option 2**: \( \frac{z_1 - z_3}{z_2 - z_4} \) is purely imaginary.
- Here, \( z_1 - z_3 \) represents one diagonal (let's say \( PR \)) and \( z_2 - z_4 \) represents the other diagonal (let's say \( QS \)).
- The diagonals intersect at right angles, hence the argument of \( \frac{z_1 - z_3}{z_2 - z_4} \) is \( \frac{\pi}{2} \), indicating it is purely imaginary.
- **Conclusion**: This option is correct.

5. **Option 3**: \( |z_1 - z_3| \neq |z_2 - z_4| \).
- Since the diagonals of a rhombus are not equal, we have \( |z_1 - z_3| \neq |z_2 - z_4| \).
- **Conclusion**: This option is correct.

6. **Option 4**: \( \frac{\text{arg}(z_1 - z_4)}{z_2 - z_4} \neq \frac{\text{arg}(z_2 - z_4)}{z_3 - z_4} \).
- The angles between the diagonals and the sides of the rhombus do not necessarily have a specific relationship that would make this statement true.
- **Conclusion**: This option is incorrect.

### Final Answer:
The correct options are:
1. \( \frac{z_1 - z_4}{z_2 - z_3} \) is purely real.
2. \( \frac{z_1 - z_3}{z_2 - z_4} \) is purely imaginary.
3. \( |z_1 - z_3| \neq |z_2 - z_4| \).