If the sides of a triangle are a, b and `sqrt(a^(2) + ab + b^(2))`, then find the greatest angle

Explanation

To find the greatest angle of the triangle with sides \( a \), \( b \), and \( c = \sqrt{a^2 + ab + b^2} \), we can use the cosine rule. The cosine rule states that for any triangle with sides \( a \), \( b \), and \( c \), and corresponding opposite angles \( A \), \( B \), and \( C \):

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( a \) and \( b \) be two sides of the triangle.
- Let \( c = \sqrt{a^2 + ab + b^2} \) be the third side.

2. **Use the cosine rule**:
- We want to find the angle \( C \) opposite to side \( c \).
- According to the cosine rule:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]

3. **Substitute \( c \)**:
- Substitute \( c = \sqrt{a^2 + ab + b^2} \) into the equation:
\[ (\sqrt{a^2 + ab + b^2})^2 = a^2 + b^2 - 2ab \cos C \]
- This simplifies to:
\[ a^2 + ab + b^2 = a^2 + b^2 - 2ab \cos C \]

4. **Rearrange the equation**:
- Subtract \( a^2 + b^2 \) from both sides:
\[ ab = -2ab \cos C \]

5. **Solve for \( \cos C \)**:
- Divide both sides by \( -2ab \) (assuming \( ab \neq 0 \)):
\[ \cos C = -\frac{1}{2} \]

6. **Find angle \( C \)**:
- The angle \( C \) for which \( \cos C = -\frac{1}{2} \) is:
\[ C = \cos^{-1}(-\frac{1}{2}) = 120^\circ \]

7. **Conclusion**:
- Therefore, the greatest angle of the triangle is \( C = 120^\circ \).