If `z_(1),z_(2), z_(3)` are vertices of an equilateral triangle with `z_(0)` its centroid, then `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=`

Options

A. `z_(0)^(2)`
B. `9z_(0)^(2)`
C. `3z_(0)^(2)`
D. `2z_(0)^(2)`

Answer

To solve the problem, we need to find the value of \( z_1^2 + z_2^2 + z_3^2 \) given that \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle and \( z_0 \) is its centroid.

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid \( z_0 \) of the triangle formed by the vertices \( z_1, z_2, z_3 \) is given by the formula:
\[ z_0 = \frac{z_1 + z_2 + z_3}{3} \]

2. **Vertices of the Equilateral Triangle**:
Let's denote the vertices of the equilateral triangle in the complex plane. For simplicity, we can place the triangle such that:
\[ z_1 = a + 0i, \quad z_2 = -\frac{a}{2} + \frac{a\sqrt{3}}{2}i, \quad z_3 = -\frac{a}{2} - \frac{a\sqrt{3}}{2}i \]
Here, \( a \) is the length of the side of the triangle.

3. **Calculating \( z_1 + z_2 + z_3 \)**:
Now, we can calculate \( z_1 + z_2 + z_3 \):
\[ z_1 + z_2 + z_3 = \left( a - \frac{a}{2} - \frac{a}{2} \right) + \left( 0 + \frac{a\sqrt{3}}{2} - \frac{a\sqrt{3}}{2} \right)i = 0 \]
Thus, \( z_0 = \frac{0}{3} = 0 \).

4. **Calculating \( z_1^2 + z_2^2 + z_3^2 \)**:
We need to compute \( z_1^2 + z_2^2 + z_3^2 \):
\[ z_1^2 = a^2 \]
\[ z_2^2 = \left(-\frac{a}{2} + \frac{a\sqrt{3}}{2}i\right)^2 = \frac{a^2}{4} - a^2\sqrt{3}i - \frac{3a^2}{4} = -\frac{2a^2}{4} + a^2\sqrt{3}i = -\frac{a^2}{2} + a^2\sqrt{3}i \]
\[ z_3^2 = \left(-\frac{a}{2} - \frac{a\sqrt{3}}{2}i\right)^2 = -\frac{a^2}{2} - a^2\sqrt{3}i \]
Now, adding these:
\[ z_1^2 + z_2^2 + z_3^2 = a^2 + \left(-\frac{a^2}{2} + a^2\sqrt{3}i\right) + \left(-\frac{a^2}{2} - a^2\sqrt{3}i\right) \]
Simplifying this gives:
\[ z_1^2 + z_2^2 + z_3^2 = a^2 - \frac{a^2}{2} - \frac{a^2}{2} + 0i = a^2 \]

5. **Relating to the Centroid**:
Since \( z_0 = 0 \), we can find \( 3z_0^2 \):
\[ 3z_0^2 = 3(0)^2 = 0 \]
Therefore, we can conclude:
\[ z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \]

### Final Result:
Thus, we have:
\[ z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \]