A chord of cicle of radius 10 cm subtends a right angle at the center . Find the area of corresponding minor segment and major segment?

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Answer

Problem:
A chord of a circle with a radius of 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment and major segment.

Solution:
To find the area of the minor segment and major segment, we need to determine the length of the chord first. Let's break down the problem step by step.

Step 1: Finding the length of the chord
Given that the chord subtends a right angle at the center, it means that it is the diameter of the circle. The diameter of a circle is twice the radius, so the length of the chord is 2 * 10 cm = 20 cm.

Step 2: Finding the area of the circle
The area of a circle is given by the formula A = πr^2, where r is the radius. In this case, the radius is 10 cm, so the area of the circle is A = π(10 cm)^2 = 100π cm^2.

Step 3: Finding the area of the minor segment
The minor segment is the region enclosed by the chord and the arc opposite to it. To find the area of the minor segment, we need to find the area of the sector formed by the central angle subtended by the chord and subtract the area of the triangle formed by the chord.

Step 3.1: Finding the area of the sector
The central angle subtended by the chord is 90 degrees. The formula to calculate the area of a sector is A = (θ/360) * πr^2, where θ is the central angle in degrees and r is the radius. Therefore, the area of the sector is A = (90/360) * π(10 cm)^2 = 25π cm^2.

Step 3.2: Finding the area of the triangle
The triangle formed by the chord is a right-angled triangle with the chord as the hypotenuse. The length of the hypotenuse is 20 cm, and the other two sides are equal to the radius, which is 10 cm. The area of a right-angled triangle is given by the formula A = (1/2) * base * height. Therefore, the area of the triangle is A = (1/2) * 10 cm * 10 cm = 50 cm^2.

Step 3.3: Calculating the area of the minor segment
To find the area of the minor segment, we subtract the area of the triangle from the area of the sector: Area of the minor segment = Area of the sector - Area of the triangle = 25π cm^2 - 50 cm^2 = (25π - 50) cm^2.

Step 4: Finding the area of the major segment
The major segment is the region enclosed by the chord and the arc opposite to it. To find the area of the major segment, we subtract the area of the minor segment from the area of the circle. Therefore, the area of the major segment is Area of the circle - Area of the minor segment = 100π cm^2 - (25π - 50) cm^2 = (100π - 25π + 50) cm^

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