A chord 10 cm long is drawn in the circle whose radius is `5 sqrt2` cm. Find the area of both the segments

Answer

To find the area of both segments created by a chord in a circle, we will follow these steps:

### Step 1: Identify the given values
- Length of the chord (AB) = 10 cm
- Radius of the circle (OA) = \(5\sqrt{2}\) cm

### Step 2: Find the distance from the center of the circle to the chord
Let O be the center of the circle, and M be the midpoint of the chord AB. Since AM = MB, we have:
- AM = MB = \( \frac{10}{2} = 5 \) cm

Using the Pythagorean theorem in triangle OMA:
\[ OA^2 = OM^2 + AM^2 \]
Substituting the values:
\[ (5\sqrt{2})^2 = OM^2 + 5^2 \]
\[ 50 = OM^2 + 25 \]
\[ OM^2 = 50 - 25 = 25 \]
\[ OM = 5 \text{ cm} \]

### Step 3: Find the angle subtended by the chord at the center
Using the cosine rule in triangle OMA:
\[ \cos(\theta) = \frac{OM}{OA} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]
Thus,
\[ \theta = 45^\circ \]
The angle subtended by the chord at the center (2θ) is:
\[ 2\theta = 90^\circ \]

### Step 4: Calculate the area of the sector OAB
The area of the sector can be calculated using the formula:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Substituting the values:
\[ \text{Area of Sector} = \frac{90}{360} \times \frac{22}{7} \times (5\sqrt{2})^2 \]
\[ = \frac{1}{4} \times \frac{22}{7} \times 50 = \frac{1100}{28} = \frac{275}{7} \text{ cm}^2 \]

### Step 5: Calculate the area of triangle OAB
The area of triangle OAB can be calculated using the formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base AB = 10 cm and height OM = 5 cm:
\[ \text{Area of Triangle} = \frac{1}{2} \times 10 \times 5 = 25 \text{ cm}^2 \]

### Step 6: Calculate the area of the minor segment
The area of the minor segment is given by:
\[ \text{Area of Minor Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
Substituting the values:
\[ \text{Area of Minor Segment} = \frac{275}{7} - 25 = \frac{275}{7} - \frac{175}{7} = \frac{100}{7} \text{ cm}^2 \]

### Step 7: Calculate the area of the major segment
The area of the major segment can be calculated as:
\[ \text{Area of Major Segment} = \text{Area of Circle} - \text{Area of Minor Segment} \]
First, we calculate the area of the circle:
\[ \text{Area of Circle} = \pi r^2 = \frac{22}{7} \times (5\sqrt{2})^2 = \frac{22}{7} \times 50 = \frac{1100}{7} \text{ cm}^2 \]
Now, substituting the values:
\[ \text{Area of Major Segment} = \frac{1100}{7} - \frac{100}{7} = \frac{1000}{7} \text{ cm}^2 \]

### Final Answers:
- Area of Minor Segment = \( \frac{100}{7} \) cm²
- Area of Major Segment = \( \frac{1000}{7} \) cm²