For `x_(1), x_(2), y_(1), y_(2) in R ` if `0 lt x_(1)lt x_(2)lt y_(1) = y_(2)` and `z_(1) = x_(1) + i y_(1), z_(2) = x_(2)+ iy_(2)` and `z_(3) = (z_(1) + z_(2))//2,`then ` z_(1) , z_(2) , z_(3)` satisfy :

Options

A. `abs(z_(1)) = abs( z_(2) ) = abs(z_(3))`
B. `abs(z_(1)) lt abs( z_(2) ) lt abs(z_(3))`
C. `abs(z_(1)) gt abs( z_(2) ) gt abs(z_(3))`
D. `abs(z_(1)) lt abs( z_(3) ) lt abs(z_(2))`

Answer

To solve the problem step by step, we will analyze the complex numbers \( z_1, z_2, \) and \( z_3 \) based on the given conditions.

### Step 1: Define the complex numbers
We have:
- \( z_1 = x_1 + i y_1 \)
- \( z_2 = x_2 + i y_2 \)
- \( z_3 = \frac{z_1 + z_2}{2} = \frac{(x_1 + x_2) + i(y_1 + y_2)}{2} \)

### Step 2: Calculate the moduli of \( z_1 \), \( z_2 \), and \( z_3 \)
The modulus of a complex number \( z = a + ib \) is given by \( |z| = \sqrt{a^2 + b^2} \).

- For \( z_1 \):
\[ |z_1| = \sqrt{x_1^2 + y_1^2} \]

- For \( z_2 \):
\[ |z_2| = \sqrt{x_2^2 + y_2^2} \]

- For \( z_3 \):
\[ |z_3| = \left| \frac{z_1 + z_2}{2} \right| = \frac{|z_1 + z_2|}{2} \]
To find \( |z_1 + z_2| \):
\[ |z_1 + z_2| = |(x_1 + x_2) + i(y_1 + y_2)| = \sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2} \]
Therefore,
\[ |z_3| = \frac{1}{2} \sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2} \]

### Step 3: Analyze the conditions
Given that \( 0 < x_1 < x_2 < y_1 = y_2 \):
- Since \( y_1 = y_2 \), we can denote \( y_1 = y_2 = y \).

Thus:
- \( |z_1| = \sqrt{x_1^2 + y^2} \)
- \( |z_2| = \sqrt{x_2^2 + y^2} \)
- \( |z_3| = \frac{1}{2} \sqrt{(x_1 + x_2)^2 + 2y^2} \)

### Step 4: Compare the moduli
Since \( x_1 < x_2 \), we know:
- \( x_1^2 < x_2^2 \)
- Therefore, \( |z_1| < |z_2| \).

Next, we will compare \( |z_3| \) with \( |z_1| \) and \( |z_2| \).

1. **Comparing \( |z_1| \) and \( |z_3| \)**:
\[ |z_3| = \frac{1}{2} \sqrt{(x_1 + x_2)^2 + 2y^2} \]
Since \( x_1 < x_2 \), it follows that:
\[ |z_3| > |z_1| \quad \text{(as the average of the real parts is greater than the smaller real part)} \]

2. **Comparing \( |z_2| \) and \( |z_3| \)**:
\[ |z_3| < |z_2| \quad \text{(as the average of the real parts is less than the larger real part)} \]

### Conclusion
From the above comparisons, we conclude:
\[ |z_1| < |z_3| < |z_2| \]

### Final Answer
Thus, the relationship that \( z_1, z_2, z_3 \) satisfy is:
\[ |z_1| < |z_3| < |z_2| \]