`|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|=|(a_1,b_1,1),(a_2,b_2,1),(a_3,b_3,1)|` then the two triangles with vertices `(x_(1), y_(1)), (x_(2), y_(2)), (x_(3), y_(3)) and (a_(1), b_(1)), (a_(2), b_(2)), (a_(3), b_(3))` are

Options

A. equal in area
B. similar
C. congruent
D. None of these

Answer

To solve the problem, we need to show that the two triangles formed by the given vertices have equal areas based on the equality of the determinants.

### Step-by-Step Solution:

1. **Understanding the Determinants**:
The area of a triangle formed by the vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the determinant:
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]
Similarly, for the triangle with vertices \((a_1, b_1)\), \((a_2, b_2)\), and \((a_3, b_3)\):
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{vmatrix} \right| \]

2. **Setting Up the Equation**:
According to the problem, we have:
\[ \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| = \left| \begin{vmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{vmatrix} \right| \]

3. **Implication of the Equality**:
Since the determinants are equal, we can denote:
\[ \Delta_1 = \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \quad \text{and} \quad \Delta_2 = \left| \begin{vmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{vmatrix} \right| \]
Thus, we have \(\Delta_1 = \Delta_2\).

4. **Calculating the Areas**:
The areas of the triangles can now be expressed as:
\[ \text{Area of Triangle 1} = \frac{1}{2} |\Delta_1| \quad \text{and} \quad \text{Area of Triangle 2} = \frac{1}{2} |\Delta_2| \]
Since \(\Delta_1 = \Delta_2\), it follows that:
\[ \text{Area of Triangle 1} = \text{Area of Triangle 2} \]

5. **Conclusion**:
Therefore, the two triangles with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) and \((a_1, b_1)\), \((a_2, b_2)\), \((a_3, b_3)\) are equal in area.

### Final Answer:
The two triangles are **equal in area**.