If the sides of a triangle are 6 cm, 10 cm and 14 cm, then what is the largest angle included by the sides ?

Options

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

Explanation

To find the largest angle included by the sides of a triangle with lengths 6 cm, 10 cm, and 14 cm, we can use the cosine rule. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( a = 6 \) cm (one side)
- Let \( b = 10 \) cm (another side)
- Let \( c = 14 \) cm (the largest side)

2. **Determine the largest angle**:
- The largest angle in a triangle is opposite the longest side. Here, the longest side is \( c = 14 \) cm, so we need to find angle \( C \) opposite side \( c \).

3. **Apply the cosine rule**:
- The cosine rule states that:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
- Rearranging this gives:
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]

4. **Substitute the values into the cosine rule**:
- Substitute \( a = 6 \), \( b = 10 \), and \( c = 14 \):
\[ \cos(C) = \frac{6^2 + 10^2 - 14^2}{2 \cdot 6 \cdot 10} \]
- Calculate \( 6^2 = 36 \), \( 10^2 = 100 \), and \( 14^2 = 196 \):
\[ \cos(C) = \frac{36 + 100 - 196}{120} \]
- Simplify the numerator:
\[ \cos(C) = \frac{136 - 196}{120} = \frac{-60}{120} = -\frac{1}{2} \]

5. **Find the angle \( C \)**:
- We know that \( \cos(C) = -\frac{1}{2} \) corresponds to:
\[ C = 120^\circ \]

### Final Answer:
The largest angle included by the sides of the triangle is \( 120^\circ \).