If the lengths of the sides of a triangle are `3, 5, 7`, then its largest angle of the triangle is

Options

A. `pi/2`
B. `(5pi)/6`
C. `(2pi)/3`
D. `(3pi)/4`

Answer

To find the largest angle of the triangle with sides of lengths 3, 5, and 7, we will use the cosine rule. The cosine rule states that for any triangle with sides \(a\), \(b\), and \(c\) opposite to angles \(A\), \(B\), and \(C\) respectively, the following relationship holds:

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

### Step-by-Step Solution:

1. **Identify the sides and the largest angle**:
- The sides of the triangle are given as \(a = 7\), \(b = 5\), and \(c = 3\).
- Since \(a\) is the largest side, angle \(A\) opposite to side \(a\) will be the largest angle.

2. **Apply the cosine rule**:
- We will calculate \(\cos A\) using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

3. **Substitute the values into the formula**:
- Plugging in the values:
\[ \cos A = \frac{5^2 + 3^2 - 7^2}{2 \cdot 5 \cdot 3} \]

4. **Calculate the squares**:
- Calculate \(5^2 = 25\), \(3^2 = 9\), and \(7^2 = 49\):
\[ \cos A = \frac{25 + 9 - 49}{2 \cdot 5 \cdot 3} \]

5. **Perform the arithmetic**:
- Simplifying the numerator:
\[ 25 + 9 - 49 = -15 \]
- And the denominator:
\[ 2 \cdot 5 \cdot 3 = 30 \]
- Therefore:
\[ \cos A = \frac{-15}{30} = -\frac{1}{2} \]

6. **Determine the angle**:
- We know that \(\cos A = -\frac{1}{2}\) corresponds to angles in the second quadrant:
\[ A = 120^\circ \]

### Conclusion:
The largest angle of the triangle is \(120^\circ\).