Suppose `z_(1), z_(2), z_(3)` represent the vertices A, B and C respectively of a `Delta ABC` with centroid at G. If the mid point of AG is the origin, then

Options

A. `z_(1) + z_(2) + z_(3) = 0`
B. `2z_(1) + z_(2) + z_(3) = 0`
C. `z_(1) + z_(2) + 4z_(3) = 0`
D. `4z_(1) + z_(2) + z_(3) = 0`

Answer

To solve the problem step by step, we need to analyze the given conditions and use the properties of centroids and midpoints in complex numbers.

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid \( G \) of triangle \( ABC \) with vertices represented by complex numbers \( z_1, z_2, z_3 \) is given by the formula:
\[ z_G = \frac{z_1 + z_2 + z_3}{3} \]

2. **Midpoint Condition**:
We are given that the midpoint of segment \( AG \) (where \( A \) is represented by \( z_1 \) and \( G \) by \( z_G \)) is the origin. The midpoint of \( AG \) can be expressed as:
\[ \text{Midpoint of } AG = \frac{z_1 + z_G}{2} \]
Setting this equal to the origin gives us:
\[ \frac{z_1 + z_G}{2} = 0 \]

3. **Solving for \( z_G \)**:
From the equation above, we can rearrange it to find:
\[ z_1 + z_G = 0 \implies z_G = -z_1 \]

4. **Substituting for \( z_G \)**:
Now, substitute the expression for \( z_G \) into the centroid formula:
\[ -z_1 = \frac{z_1 + z_2 + z_3}{3} \]

5. **Multiplying through by 3**:
To eliminate the fraction, multiply both sides by 3:
\[ -3z_1 = z_1 + z_2 + z_3 \]

6. **Rearranging the Equation**:
Bringing all terms involving \( z_1 \) to one side gives:
\[ -3z_1 - z_1 = z_2 + z_3 \implies -4z_1 = z_2 + z_3 \]

7. **Final Relation**:
Thus, we can express the relationship among the vertices as:
\[ 4z_1 + z_2 + z_3 = 0 \]

### Conclusion:
The relationship between the vertices \( z_1, z_2, z_3 \) is given by:
\[ 4z_1 + z_2 + z_3 = 0 \]