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

Options

A. `90^(@)`
B. `150^(@)`
C. `120^(@)`
D. `135^(@)`

Answer

To find the largest angle of a triangle with sides of lengths 3, 5, and 7, we can 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 \(a = 3\), \(b = 5\), and \(c = 7\).
- The largest side is \(c = 7\), so the largest angle \(A\) is opposite this side.

2. **Apply the cosine rule**:
- We need to find \(\cos A\):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Here, \(b = 5\), \(c = 7\), and \(a = 3\).

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

4. **Calculate the squares**:
- \(5^2 = 25\)
- \(7^2 = 49\)
- \(3^2 = 9\)

5. **Substitute the squared values**:
\[ \cos A = \frac{25 + 49 - 9}{2 \cdot 5 \cdot 7} \]

6. **Simplify the numerator**:
\[ 25 + 49 - 9 = 65 \]
So,
\[ \cos A = \frac{65}{2 \cdot 5 \cdot 7} \]

7. **Calculate the denominator**:
\[ 2 \cdot 5 \cdot 7 = 70 \]
Thus,
\[ \cos A = \frac{65}{70} = \frac{13}{14} \]

8. **Find angle A using the cosine inverse**:
- To find angle \(A\), we take the inverse cosine:
\[ A = \cos^{-1}\left(\frac{13}{14}\right) \]

9. **Determine the largest angle**:
- Since \(A\) is opposite the largest side, we can also find the angle using the sine rule or directly from the cosine value. However, we can also determine that the largest angle in a triangle with sides 3, 5, and 7 is \(A\).

10. **Conclusion**:
- The largest angle \(A\) is approximately \(120^\circ\).