A chord xy of a circle of radiuns 14cm subtends angle of 60 degree at the centre. find the area of major segment.
The area of the minor segment is
1/2 r^2 (θ-sinθ)
So, plug in your values and subtract that from the area of the circle, πr^2
To find the area of the major segment, we need to first find the area of the corresponding sector and then subtract the area of the triangle formed by the chord.
Step 1: Find the length of the chord.
In a circle, the length of a chord can be found using the formula:
Length of chord = 2 * radius * sin(angle/2)
Given that the radius is 14 cm and the angle is 60 degrees, we can substitute these values into the formula:
Length of chord = 2 * 14 cm * sin(60/2)
Length of chord = 2 * 14 cm * sin(30)
Length of chord = 2 * 14 cm * 0.5 (since sin(30) = 0.5)
Length of chord = 14 cm
Step 2: Find the area of the sector.
The area of a sector can be found using the formula:
Area of sector = (angle/360) * π * radius^2
Given that the angle is 60 degrees and the radius is 14 cm, we can substitute these values into the formula:
Area of sector = (60/360) * π * (14 cm)^2
Area of sector = (1/6) * π * (14 cm)^2
Area of sector ≈ 129.6 cm^2
Step 3: Find the area of the triangle.
The area of a triangle can be found using the formula:
Area of triangle = (base * height) / 2
In this case, the base of the triangle is the length of the chord, which we found to be 14 cm. The height can be found by drawing a perpendicular from the center of the circle to the chord, creating a right triangle. The height is the distance from the center of the circle to the midpoint of the chord. Since the angle of the sector is 60 degrees, this forms an equilateral triangle, and the height can be found using the formula:
Height = (√3/2) * length of side
In this case, the length of the side is equal to the length of the chord, which is 14 cm.
Height = (√3/2) * 14 cm
Height ≈ 12.1 cm
Now we can find the area of the triangle:
Area of triangle = (14 cm * 12.1 cm) / 2
Area of triangle ≈ 84.7 cm^2
Step 4: Find the area of the major segment.
The area of the major segment is the difference between the area of the sector and the area of the triangle:
Area of major segment = Area of sector - Area of triangle
Area of major segment ≈ 129.6 cm^2 - 84.7 cm^2
Area of major segment ≈ 44.9 cm^2
Therefore, the area of the major segment is approximately 44.9 cm^2.