If two sides and included angle of a triangle are respectively `3+sqrt(3),3-sqrt(3)` and `60^(@)`, then the third sides is

Options

A. `2 sqrt(2)`
B. `4sqrt(2)`
C. `3sqrt(2)`
D. none of these

Explanation

To solve the given problem, we will use the Law of Cosines. The question states that we have a triangle with two sides and the included angle. The sides are \( a = 3 + \sqrt{3} \), \( b = 3 - \sqrt{3} \), and the included angle \( C = 60^\circ \). We need to find the length of the third side \( c \).

### Step-by-Step Solution:

1. **Identify the given values**:
- \( a = 3 + \sqrt{3} \)
- \( b = 3 - \sqrt{3} \)
- \( C = 60^\circ \)

2. **Apply the Law of Cosines**:
The Law of Cosines states that:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
Substituting the values we have:
\[ c^2 = (3 + \sqrt{3})^2 + (3 - \sqrt{3})^2 - 2(3 + \sqrt{3})(3 - \sqrt{3}) \cos(60^\circ) \]

3. **Calculate \( a^2 \) and \( b^2 \)**:
\[ a^2 = (3 + \sqrt{3})^2 = 9 + 6\sqrt{3} + 3 = 12 + 6\sqrt{3} \]
\[ b^2 = (3 - \sqrt{3})^2 = 9 - 6\sqrt{3} + 3 = 12 - 6\sqrt{3} \]

4. **Calculate \( 2ab \cos(60^\circ) \)**:
Since \( \cos(60^\circ) = \frac{1}{2} \):
\[ 2ab \cos(60^\circ) = 2(3 + \sqrt{3})(3 - \sqrt{3}) \cdot \frac{1}{2} \]
First, calculate \( ab \):
\[ ab = (3 + \sqrt{3})(3 - \sqrt{3}) = 9 - 3 = 6 \]
Therefore:
\[ 2ab \cos(60^\circ) = 6 \]

5. **Substitute back into the equation**:
\[ c^2 = (12 + 6\sqrt{3}) + (12 - 6\sqrt{3}) - 6 \]
Simplifying:
\[ c^2 = 12 + 6\sqrt{3} + 12 - 6\sqrt{3} - 6 = 18 \]

6. **Find \( c \)**:
\[ c = \sqrt{18} = 3\sqrt{2} \]

### Final Answer:
The length of the third side \( c \) is \( 3\sqrt{2} \).