Assertion (A) : If 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))` satisfy the relation
`|{:(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 triangles are congruent
Reason (R): For the given triangles satisfying the above relation implies that the triangles have equal area.

Options

A. A. Both A and R are individually true and R is the correct explanation of A
B. B. Both A and R are individually true but R is not correct explanation of A
C. C. A is true but R is false
D. D. A is false but R is true

Answer

To solve the problem, we need to analyze the assertion and the reason provided in the question step by step.

### Step 1: Understanding the Assertion
The assertion states that if two triangles with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) and \((a_1, b_1)\), \((a_2, b_2)\), \((a_3, b_3)\) satisfy the relation
\[ |{(x_1, y_1, 1), (x_2, y_2, 1), (x_3, y_3, 1)}| = |{(a_1, b_1, 1), (a_2, b_2, 1), (a_3, b_3, 1)}| \]
then the triangles are congruent.

### Step 2: Understanding the Reason
The reason states that the above relation implies that the triangles have equal area.

### Step 3: Analyzing the Determinant
The determinant of the matrix formed by the vertices of the triangles gives us a value related to the area of the triangle. The area \(A\) of a triangle formed by points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]
Thus, the assertion is correct if the areas of both triangles are equal.

### Step 4: Conclusion
Since the assertion states that the triangles are congruent based on the equality of their areas, and the reason correctly explains that the given relation implies equal areas, we conclude that both the assertion and the reason are correct.

### Final Answer
Both the assertion and the reason are individually correct, and the reason is the correct explanation of the assertion.