A chord subtends an angle `120^(@)` at the centre of the circle of radius 1 unit . What is the length of the chord?

Options

A. `sqrt(2)-1` units
B. `sqrt(3)-1` units
C. `sqrt(2)` units
D. `sqrt(3)` units

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 1 unit, we can use the following steps:

### Step 1: Understand the Geometry
We have a circle with center \(O\) and a chord \(AB\) that subtends an angle of \(120^\circ\) at the center. The points \(A\) and \(B\) are on the circumference of the circle, and the radius \(OA\) and \(OB\) are both equal to 1 unit.

### Step 2: Use the Cosine Rule
In triangle \(OAB\), we can apply the cosine rule, which states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where \(C\) is the angle opposite side \(c\), and \(a\) and \(b\) are the lengths of the other two sides.

In our case:
- \(a = OA = 1\)
- \(b = OB = 1\)
- \(C = 120^\circ\)
- \(c = AB\) (the length of the chord we want to find)

### Step 3: Substitute Values into the Cosine Rule
Substituting the values into the cosine rule:
\[ AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(120^\circ) \]
\[ AB^2 = 1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(120^\circ) \]

### Step 4: Calculate \(\cos(120^\circ)\)
We know that:
\[ \cos(120^\circ) = -\frac{1}{2} \]
Now substituting this value:
\[ AB^2 = 1 + 1 - 2 \cdot 1 \cdot 1 \cdot \left(-\frac{1}{2}\right) \]
\[ AB^2 = 1 + 1 + 1 = 3 \]

### Step 5: Find the Length of the Chord \(AB\)
Taking the square root of both sides:
\[ AB = \sqrt{3} \]

### Conclusion
The length of the chord \(AB\) is \(\sqrt{3}\) units.