In a `Delta ABC, angle A=120^(@), AB =6 cm, AC=8 cm` find the l,ength of side BC.

Answer

To find the length of side BC in triangle ABC where angle A = 120°, AB = 6 cm, and AC = 8 cm, we can use the Law of Cosines. The Law of Cosines states that for any triangle ABC:

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

Where:
- \( c \) is the side opposite angle C,
- \( a \) and \( b \) are the other two sides,
- \( C \) is the angle opposite side \( c \).

In our case:
- Let \( a = AC = 8 \) cm,
- Let \( b = AB = 6 \) cm,
- Let \( C = \angle A = 120° \),
- Let \( c = BC \) (the side we want to find).

### Step 1: Substitute the values into the Law of Cosines formula.

\[ BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(A) \]

Substituting the known values:

\[ BC^2 = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cdot \cos(120°) \]

### Step 2: Calculate \( \cos(120°) \).

The cosine of 120° is:

\[ \cos(120°) = -\frac{1}{2} \]

### Step 3: Substitute \( \cos(120°) \) into the equation.

\[ BC^2 = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cdot \left(-\frac{1}{2}\right) \]

### Step 4: Calculate \( 6^2 \) and \( 8^2 \).

\[ 6^2 = 36 \]
\[ 8^2 = 64 \]

### Step 5: Substitute these values back into the equation.

\[ BC^2 = 36 + 64 - 2 \cdot 6 \cdot 8 \cdot \left(-\frac{1}{2}\right) \]

### Step 6: Simplify the equation.

\[ BC^2 = 36 + 64 + 2 \cdot 6 \cdot 8 \cdot \frac{1}{2} \]
\[ BC^2 = 36 + 64 + 48 \]

### Step 7: Add the values together.

\[ BC^2 = 148 \]

### Step 8: Take the square root to find BC.

\[ BC = \sqrt{148} \]
\[ BC = \sqrt{4 \cdot 37} \]
\[ BC = 2\sqrt{37} \]

### Final Answer:
The length of side BC is \( 2\sqrt{37} \) cm.

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