In a `Delta ABC, a,b,A ` are given and `c_(1), c_(2)` are two values of the third side c. The sum of the areas two triangles with sides a,b,`c_(1) and a,b,c_(2)` is

Options

A. `1/2a ^(2) sin 2A`
B. `1/2b ^(2)sin 2A`
C. `b ^(2)sin 2A`
D. None of these

Answer

To solve the problem, we need to find the sum of the areas of two triangles with sides \( a, b, c_1 \) and \( a, b, c_2 \). We will use the sine formula for the area of a triangle, which states that the area \( A \) of a triangle with sides \( a \) and \( b \) and included angle \( A \) is given by:

\[ \text{Area} = \frac{1}{2}ab \sin A \]

### Step-by-Step Solution:

1. **Identify the Given Values**:
We are given the sides \( a \), \( b \), and the angle \( A \) of triangle \( ABC \). We also have two values for the third side \( c \): \( c_1 \) and \( c_2 \).

2. **Use the Sine Formula for Area**:
The area of triangle \( ABC \) with sides \( a, b, c_1 \) is given by:
\[ \text{Area}_1 = \frac{1}{2} b c_1 \sin A \]
The area of triangle \( ABC \) with sides \( a, b, c_2 \) is given by:
\[ \text{Area}_2 = \frac{1}{2} b c_2 \sin A \]

3. **Sum of the Areas**:
The total area of both triangles is:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = \frac{1}{2} b c_1 \sin A + \frac{1}{2} b c_2 \sin A \]
Factoring out the common terms gives:
\[ \text{Total Area} = \frac{1}{2} b \sin A (c_1 + c_2) \]

4. **Using the Roots of the Quadratic Equation**:
Since \( c_1 \) and \( c_2 \) are the roots of the quadratic equation derived from the cosine rule, we know:
\[ c_1 + c_2 = 2b \cos A \]
Therefore, we can substitute this into the total area expression:
\[ \text{Total Area} = \frac{1}{2} b \sin A (2b \cos A) \]

5. **Simplifying the Expression**:
This simplifies to:
\[ \text{Total Area} = b^2 \sin A \cos A \]

6. **Using the Double Angle Identity**:
We can use the double angle identity for sine:
\[ \sin 2A = 2 \sin A \cos A \]
Thus, we have:
\[ \text{Total Area} = \frac{1}{2} b^2 \sin 2A \]

### Final Result:
The sum of the areas of the two triangles is:
\[ \text{Total Area} = \frac{1}{2} b^2 \sin 2A \]