Suppose `z_(1), z_(2), z_(3)` are vertices of an equilateral triangle whose circumcentre -3 + 4i, then `|z_(1) + z_(2) + z_(3)|` is equal to

Options

A. 5
B. `10 sqrt(3)`
C. 15
D. `15 sqrt(3)`

Answer

To solve the problem, we need to find the value of \( |z_1 + z_2 + z_3| \) given that the circumcenter of the equilateral triangle formed by the vertices \( z_1, z_2, z_3 \) is \( -3 + 4i \).

### Step-by-Step Solution:

1. **Understanding the Circumcenter**:
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. For an equilateral triangle, the circumcenter is also the centroid.

2. **Finding the Centroid**:
The centroid \( G \) of a triangle with vertices \( z_1, z_2, z_3 \) can be calculated using the formula:
\[ G = \frac{z_1 + z_2 + z_3}{3} \]
Since the circumcenter is given as \( -3 + 4i \), we can set:
\[ \frac{z_1 + z_2 + z_3}{3} = -3 + 4i \]

3. **Multiplying by 3**:
To eliminate the fraction, multiply both sides by 3:
\[ z_1 + z_2 + z_3 = 3(-3 + 4i) \]

4. **Calculating the Right Side**:
Calculate the right side:
\[ z_1 + z_2 + z_3 = -9 + 12i \]

5. **Finding the Modulus**:
Now, we need to find the modulus of \( z_1 + z_2 + z_3 \):
\[ |z_1 + z_2 + z_3| = |-9 + 12i| \]
The modulus of a complex number \( a + bi \) is given by:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
Here, \( a = -9 \) and \( b = 12 \):
\[ |z_1 + z_2 + z_3| = \sqrt{(-9)^2 + (12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \]

### Final Answer:
Thus, the value of \( |z_1 + z_2 + z_3| \) is \( 15 \).

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