If `z_(1),z_(2),z_(3)` are the vertices of an equilational triangle ABC such that `|z_(1)-i|=|z_(2)- i| = |z_(3)-i|,` then `|z_(1)+z_(2)+z_(3)|` equals to

Options

A. `3sqrt(3)`
B. `sqrt(3)`
C. 3
D. `(1)/(3sqrt(3)`

Explanation

To solve the problem, we need to find the value of \( |z_1 + z_2 + z_3| \) given that \( |z_1 - i| = |z_2 - i| = |z_3 - i| \). This indicates that the points \( z_1, z_2, z_3 \) are equidistant from the point \( i \) in the complex plane.

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \( |z_1 - i| = |z_2 - i| = |z_3 - i| \) implies that the points \( z_1, z_2, z_3 \) lie on a circle centered at \( i \) (which is the point \( (0, 1) \) in the complex plane) with some radius \( r \).

**Hint**: Recognize that the points being equidistant from a single point means they lie on a circle.

2. **Finding the Centroid**:
The centroid \( G \) of the triangle formed by the points \( z_1, z_2, z_3 \) is given by:
\[ G = \frac{z_1 + z_2 + z_3}{3} \]
Thus, we can express \( z_1 + z_2 + z_3 \) as:
\[ z_1 + z_2 + z_3 = 3G \]

**Hint**: The centroid of a triangle is the average of its vertices.

3. **Position of the Centroid**:
Since \( z_1, z_2, z_3 \) are vertices of an equilateral triangle, the centroid \( G \) will also lie on the line that connects the center of the circle (which is \( i \)) to the centroid of the triangle.

**Hint**: The centroid of an equilateral triangle is also the center of the circle circumscribing it.

4. **Calculating the Modulus**:
The distance from the center \( i \) to any vertex \( z_k \) (where \( k = 1, 2, 3 \)) is the radius \( r \). The centroid \( G \) will be at a distance of \( \frac{r}{\sqrt{3}} \) from \( i \) (since the centroid divides the median in a 2:1 ratio).

Therefore, the modulus of \( z_1 + z_2 + z_3 \) is:
\[ |z_1 + z_2 + z_3| = |3G| = 3|G| \]

**Hint**: Remember the relationship between the centroid and the radius of the circumcircle in an equilateral triangle.

5. **Final Calculation**:
Since \( G \) is at a distance \( r \) from \( i \) and the triangle is equilateral, the distance \( |G - i| \) can be determined to be \( \frac{r}{\sqrt{3}} \). However, since the points are symmetric about \( i \), we can conclude that:
\[ |z_1 + z_2 + z_3| = 3 \cdot |G| = 3 \cdot r \]
Since \( r \) is the radius of the circle centered at \( i \), we can conclude that:
\[ |z_1 + z_2 + z_3| = 3 \]

**Hint**: The symmetry of the equilateral triangle simplifies the calculation of the centroid and its distance from the center.

### Conclusion:
Thus, the final answer is:
\[ |z_1 + z_2 + z_3| = 3 \]