An equilateral triangle has each side to a. If the coordinates of its vertices are `(x_(1), y_(1)), (x_(2), y_(2))` and `(x_(3), y_(3))` then the square of the determinat `|(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|` equals

Options

A. `3a^(4)`
B. `(3a^(4))/(2)`
C. `(3)/(4)a^(4)`
D. `(3)/(8)a^(4)`

Answer

To find the square of the determinant for the vertices of an equilateral triangle with each side of length \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Area of an Equilateral Triangle**:
The area \( A \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

2. **Using the Determinant to Find the Area**:
The area of a triangle formed by points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can also be expressed using the determinant:
\[ A = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]

3. **Setting Up the Equation**:
Equating the two expressions for the area, we have:
\[ \frac{\sqrt{3}}{4} a^2 = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]

4. **Simplifying the Equation**:
Multiply both sides by 2 to eliminate the fraction:
\[ \frac{\sqrt{3}}{2} a^2 = \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]

5. **Squaring Both Sides**:
To find the square of the determinant, we square both sides:
\[ \left( \frac{\sqrt{3}}{2} a^2 \right)^2 = \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|^2 \]
This simplifies to:
\[ \frac{3}{4} a^4 = \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|^2 \]

6. **Final Result**:
Therefore, the square of the determinant is:
\[ \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|^2 = \frac{3}{4} a^4 \]

### Conclusion:
The correct answer is \( \frac{3a^4}{4} \).