Find the length of the chord subtending an angle of `120^(@)` at the centre of the circle whose radius is 4 cm.

Answer

To find the length of the chord that subtends an angle of \(120^\circ\) at the center of a circle with a radius of 4 cm, we can use the formula for the length of a chord:

\[ L = 2r \sin\left(\frac{\theta}{2}\right) \]

where:
- \(L\) is the length of the chord,
- \(r\) is the radius of the circle,
- \(\theta\) is the angle subtended at the center in degrees.

### Step-by-step Solution:

1. **Identify the given values:**
- Radius \(r = 4 \, \text{cm}\)
- Angle \(\theta = 120^\circ\)

2. **Calculate \(\frac{\theta}{2}\):**
\[ \frac{\theta}{2} = \frac{120^\circ}{2} = 60^\circ \]

3. **Find \(\sin\left(60^\circ\right)\):**
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

4. **Substitute the values into the chord length formula:**
\[ L = 2 \times 4 \times \sin(60^\circ) \]
\[ L = 2 \times 4 \times \frac{\sqrt{3}}{2} \]

5. **Simplify the expression:**
\[ L = 4 \times \sqrt{3} \]

6. **Calculate the numerical value (if needed):**
\[ L \approx 4 \times 1.732 = 6.928 \, \text{cm} \]

### Final Answer:
The length of the chord is \(4\sqrt{3} \, \text{cm}\) or approximately \(6.93 \, \text{cm}\).