Understanding the Problem
To find the area of the minor segment formed by a chord that subtends a right angle at the center of a circle with a radius of 21 cm, we need to approach the problem step by step.
Step 1: Calculate the Area of the Sector
- The angle subtended at the center is 90 degrees (or π/2 radians).
- The area of the entire circle is given by the formula A = πr². Here, r = 21 cm.
- Therefore, the area of the sector with a 90-degree angle is:
- Area of sector = (θ/360) * πr² = (90/360) * π(21)².
Step 2: Calculate the Area of the Triangle
- The triangle formed by the radius and the chord is a right-angled triangle.
- Both legs of the triangle are equal to the radius (21 cm).
- Area of the triangle = (1/2) * base * height = (1/2) * 21 * 21.
Step 3: Calculate the Area of the Minor Segment
- The area of the minor segment can be found by subtracting the area of the triangle from the area of the sector:
- Area of minor segment = Area of sector - Area of triangle.
Final Calculation
- After performing the calculations, you will find:
- Area of sector = (1/4) * π * 21².
- Area of triangle = (1/2) * 21 * 21.
- Finally, Area of minor segment = Area of sector - Area of triangle.
Conclusion
The area of the corresponding minor segment is the difference between the sector and triangle areas, providing a comprehensive understanding of the geometrical relationship within the circle.