If the complex number `z_(1)` and `z_(2)` and the origin form an equilateral triangle, then `z_(1)^(2) + z_(2)^(2)` is equal to

Options

A. A. `z_(1) z_(2)`
B. B. `z_(1) bar(z)_(2)`
C. C. `bar(z)_(2)z_(1)`
D. D. `|z_(1)|^(2) =|z_(2)|^(2)`

Answer

To solve the problem, we need to find the value of \( z_1^2 + z_2^2 \) given that the complex numbers \( z_1 \), \( z_2 \), and the origin (0) form an equilateral triangle.

### Step-by-Step Solution:

1. **Understanding the properties of an equilateral triangle**:
For any three points \( z_1 \), \( z_2 \), and \( z_3 \) (where \( z_3 \) is the origin in this case), the following relationship holds:
\[ z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1 \]

2. **Substituting the value of \( z_3 \)**:
Since \( z_3 = 0 \), we can substitute this into the equation:
\[ z_1^2 + z_2^2 + 0^2 = z_1 z_2 + z_2 \cdot 0 + 0 \cdot z_1 \]
This simplifies to:
\[ z_1^2 + z_2^2 = z_1 z_2 \]

3. **Final expression**:
Therefore, we conclude that:
\[ z_1^2 + z_2^2 = z_1 z_2 \]

### Conclusion:
The value of \( z_1^2 + z_2^2 \) is equal to \( z_1 z_2 \).