An equilateral triangle has each of its sides of length 4 cm. If `(x_(r),y_(r))` (r=1,2,3) are its vertices the value of `|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^2`

Options

A. 192
B. 768
C. 1024
D. 128

Answer

To solve the problem, we need to find the value of the determinant \( |(x_1, y_1, 1), (x_2, y_2, 1), (x_3, y_3, 1)|^2 \) for the vertices of an equilateral triangle with each side of length 4 cm.

### Step 1: Determine the vertices of the equilateral triangle

Let's place the equilateral triangle in the coordinate system. We can choose the vertices as follows:
- Vertex \( A \) at \( (0, 0) \)
- Vertex \( B \) at \( (4, 0) \)
- Vertex \( C \) at \( (2, 2\sqrt{3}) \)

This configuration ensures that the triangle is equilateral with each side measuring 4 cm.

### Step 2: Set the coordinates

From the chosen vertices, we have:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (4, 0) \)
- \( (x_3, y_3) = (2, 2\sqrt{3}) \)

### Step 3: Write the determinant

The determinant we need to calculate is:
\[ D = \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \begin{vmatrix} 0 & 0 & 1 \\ 4 & 0 & 1 \\ 2 & 2\sqrt{3} & 1 \end{vmatrix} \]

### Step 4: Calculate the determinant

Using the determinant formula for a 3x3 matrix:
\[ D = x_1(y_2 \cdot 1 - y_3 \cdot 1) - y_1(x_2 \cdot 1 - x_3 \cdot 1) + 1(x_2y_3 - x_3y_2) \]
Substituting the values:
\[ D = 0(0 - 2\sqrt{3}) - 0(4 - 2) + 1(4 \cdot 2\sqrt{3} - 2 \cdot 0) \]
\[ D = 0 + 0 + 8\sqrt{3} \]
Thus, \( D = 8\sqrt{3} \).

### Step 5: Square the determinant

Now, we need to find \( |D|^2 \):
\[ |D|^2 = (8\sqrt{3})^2 = 64 \cdot 3 = 192 \]

### Final Answer

The value of \( |(x_1, y_1, 1), (x_2, y_2, 1), (x_3, y_3, 1)|^2 \) is \( 192 \).