A chord of a circle subtends an angle of `60^(@)` at the centre. If the length of the chord is 100 cm, 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 given a chord that subtends an angle of \(60^\circ\) at the center and has a length of 100 cm, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with center \(O\) and a chord \(AB\) that subtends an angle of \(60^\circ\) at the center. The length of the chord \(AB\) is given as 100 cm.

### Step 2: Find the Radius of the Circle
To find the radius \(r\) of the circle, we can use the formula for the length of a chord:
\[ AB = 2r \sin\left(\frac{\theta}{2}\right) \]
where \(\theta\) is the angle subtended at the center. Here, \(\theta = 60^\circ\).

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

### Step 3: Calculate the Area of the Sector
The area \(A\) of the sector formed by the angle \(60^\circ\) can be calculated using the formula:
\[ A = \frac{\theta}{360^\circ} \cdot \pi r^2 \]
Substituting the values:
\[ A = \frac{60}{360} \cdot \pi \cdot (100)^2 \]
\[ A = \frac{1}{6} \cdot \pi \cdot 10000 \]
\[ A = \frac{10000\pi}{6} \approx 5235.99 \text{ cm}^2 \]

### Step 4: Calculate the Area of Triangle \(OAB\)
The area of triangle \(OAB\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \cdot AB \cdot h \]
where \(h\) is the height from \(O\) to the chord \(AB\).

Since triangle \(OAB\) is an equilateral triangle (as we derived earlier), we can also use the formula for the area of an equilateral triangle:
\[ \text{Area} = \frac{\sqrt{3}}{4} \cdot a^2 \]
where \(a\) is the length of the side (which is equal to the length of the chord):
\[ \text{Area} = \frac{\sqrt{3}}{4} \cdot (100)^2 \]
\[ \text{Area} = \frac{\sqrt{3}}{4} \cdot 10000 \approx 4330.13 \text{ cm}^2 \]

### Step 5: Calculate the Area of the Major Segment
The area of the major segment is given by:
\[ \text{Area of Major Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
\[ \text{Area of Major Segment} = 5235.99 - 4330.13 \approx 905.86 \text{ cm}^2 \]

### Step 6: Final Calculation
To find the area of the major segment, we need to subtract the area of the triangle from the area of the sector:
\[ \text{Area of Major Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
\[ \text{Area of Major Segment} = 5235.99 - 4330.13 \approx 905.86 \text{ cm}^2 \]

### Conclusion
Thus, the area of the major segment is approximately \(905.86 \text{ cm}^2\).