A chord of a circle subtends an angle of `60^(@)` at the centre. If the length of the chord is 100 cm, then find the area of the major segment.

Options

A. `30720.5 cm^(2)`
B. `31021.42 cm^(2)`
C. `30391.7 cm^(2)`
D. `30520.61 cm^(2)`

Answer

To find the area of the major segment of a circle subtended by a chord of length 100 cm at an angle of \(60^\circ\) at the center, we can follow these steps:

### Step 1: Find the radius of the circle
Given that the chord subtends an angle of \(60^\circ\) at the center and the length of the chord (AB) is 100 cm, we can use the formula for the length of the chord:
\[ AB = 2r \sin\left(\frac{\theta}{2}\right) \]
where \( \theta \) is the angle subtended at the center in degrees.

Substituting the values:
\[ 100 = 2r \sin\left(\frac{60}{2}\right) = 2r \sin(30) \]
Since \( \sin(30) = \frac{1}{2} \):
\[ 100 = 2r \cdot \frac{1}{2} = r \]
Thus, the radius \( r = 100 \) cm.

### Step 2: Calculate the area of the sector
The area of the sector formed by the angle \( \theta \) is given by:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
Substituting \( \theta = 60^\circ \) and \( r = 100 \) cm:
\[ \text{Area of sector} = \frac{60}{360} \times \pi \times (100)^2 \]
\[ = \frac{1}{6} \times \pi \times 10000 \]
\[ = \frac{10000\pi}{6} \approx 5233.33 \text{ cm}^2 \quad (\text{using } \pi \approx 3.14) \]

### Step 3: Calculate the area of triangle OAB
Since triangle OAB is an equilateral triangle (as both OA and OB are radii and the angle AOB is \(60^\circ\)), we can use the formula for the area of an equilateral triangle:
\[ \text{Area of triangle} = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the length of the side (which is equal to the radius):
\[ \text{Area of triangle} = \frac{\sqrt{3}}{4} (100)^2 = \frac{\sqrt{3}}{4} \times 10000 \]
\[ \approx \frac{1732}{4} \approx 433.00 \text{ cm}^2 \]

### Step 4: Calculate the area of the minor segment
The area of the minor segment is given by:
\[ \text{Area of minor segment} = \text{Area of sector} - \text{Area of triangle} \]
\[ = 5233.33 - 433.00 \approx 4800.33 \text{ cm}^2 \]

### Step 5: Calculate the area of the major segment
The area of the major segment is given by:
\[ \text{Area of major segment} = \text{Area of circle} - \text{Area of minor segment} \]
The area of the circle is:
\[ \text{Area of circle} = \pi r^2 = \pi \times (100)^2 = 31400 \text{ cm}^2 \]
Thus,
\[ \text{Area of major segment} = 31400 - 4800.33 \approx 26600 \text{ cm}^2 \]

### Final Answer
The area of the major segment is approximately \( 26600 \text{ cm}^2 \).