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 in the triangle with sides \( a \), \( b \), and \( \sqrt{a^2 + ab + b^2} \), we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- Side 1: \( a \)
- Side 2: \( b \)
- Side 3: \( C = \sqrt{a^2 + ab + b^2} \)

### Step 2: Determine the greatest side
To find the greatest angle, we need to identify the greatest side. We compare \( a \), \( b \), and \( C \):
- Since \( C = \sqrt{a^2 + ab + b^2} \), we can see that \( C \) is greater than both \( a \) and \( b \) because:
\[ C^2 = a^2 + ab + b^2 > a^2 \quad \text{and} \quad C^2 = a^2 + ab + b^2 > b^2 \]
Thus, the greatest side is \( C \).

### Step 3: Use the cosine rule to find the greatest angle
The angle opposite the greatest side \( C \) is denoted as \( \angle C \). Using the cosine rule:
\[ \cos C = \frac{a^2 + b^2 - C^2}{2ab} \]
Substituting \( C \) into the equation:
\[ \cos C = \frac{a^2 + b^2 - (a^2 + ab + b^2)}{2ab} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ \cos C = \frac{a^2 + b^2 - a^2 - ab - b^2}{2ab} = \frac{-ab}{2ab} \]
This simplifies to:
\[ \cos C = -\frac{1}{2} \]

### Step 5: Find the angle \( C \)
To find the angle \( C \), we take the inverse cosine:
\[ C = \cos^{-1}\left(-\frac{1}{2}\right) \]
This corresponds to:
\[ C = 120^\circ \]

### Final Answer
Thus, the greatest angle in the triangle is \( \angle C = 120^\circ \).

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