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 cm^2`
C. `30391.7 cm^2`
D. `26190.47 cm^2`

Answer

To find the area of the major segment of a circle given that a chord subtends an angle of \(60^\circ\) at the center and the length of the chord is 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 points \(A\) and \(B\) are on the circumference of the circle.

### Step 2: Find the Radius
Since the chord \(AB\) subtends an angle of \(60^\circ\) at the center, we can use the properties of triangles to find the radius \(r\).

In triangle \(OAB\), we know:
- \(OA = OB = r\) (radii of the circle)
- \(AB = 100 \text{ cm}\) (length of the chord)

Using 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.

Substituting the known 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 \(r\) is \(100 \text{ cm}\).

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

### Step 4: Calculate the Area of the Triangle
The area \(A_T\) of triangle \(OAB\) can be calculated using the formula:
\[ A_T = \frac{1}{2} \cdot OA \cdot OB \cdot \sin(\theta) \]
Substituting the known values:
\[ A_T = \frac{1}{2} \cdot 100 \cdot 100 \cdot \sin(60^\circ) \]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ A_T = \frac{1}{2} \cdot 100 \cdot 100 \cdot \frac{\sqrt{3}}{2} \]
\[ A_T = 2500\sqrt{3} \approx 4330.13 \text{ cm}^2 \]

### Step 5: Calculate the Area of the Major Segment
The area of the major segment is the area of the sector minus the area of the triangle:
\[ A_{major\ segment} = A - A_T \]
\[ A_{major\ segment} = \frac{10000\pi}{6} - 2500\sqrt{3} \]
Substituting the approximate values of \(\pi \approx 3.14\) and \(\sqrt{3} \approx 1.732\):
\[ A_{major\ segment} \approx \frac{10000 \cdot 3.14}{6} - 2500 \cdot 1.732 \]
\[ A_{major\ segment} \approx 5235.99 - 4330.13 \approx 905.86 \text{ cm}^2 \]

### Final Answer
The area of the major segment is approximately \(905.86 \text{ cm}^2\).