If `|x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_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 equal to area (b) similar congruent (d) none of these

Options

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

Explanation

To solve the problem, we need to analyze the given determinant equations that represent the areas of two triangles formed by their respective vertices. Let's go through the steps:

### Step-by-Step Solution:

1. **Understanding the Determinant**:
The expression given in the problem is:
\[ |x_1 y_1 1 \quad x_2 y_2 1 \quad x_3 y_3 1| = |a_1 b_1 1 \quad a_2 b_2 1 \quad a_3 b_3 1| \]
This determinant represents twice the area of the triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) and the points \((a_1, b_1)\), \((a_2, b_2)\), \((a_3, b_3)\).

2. **Multiplying by \(\frac{1}{2}\)**:
To find the area of the triangles, we can multiply both sides of the equation by \(\frac{1}{2}\):
\[ \frac{1}{2} |x_1 y_1 1 \quad x_2 y_2 1 \quad x_3 y_3 1| = \frac{1}{2} |a_1 b_1 1 \quad a_2 b_2 1 \quad a_3 b_3 1| \]
This simplifies to:
\[ \text{Area of triangle 1} = \text{Area of triangle 2} \]

3. **Conclusion**:
Since the areas of both triangles are equal, we conclude that the triangles have the same area.

4. **Identifying the Correct Option**:
The options provided are:
- (a) equal to area
- (b) similar
- (c) congruent
- (d) none of these

Since we have established that the areas are equal, the correct option is (a) equal to area.

### Final Answer:
The triangles are equal in area. Therefore, the answer is **(a) equal to area**.

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