A chord of radius 12cm substend an angle of 150°at the centre of the circle.find the area of the segment bounded by the chord and the Minor Arc
The sector has area 1/2 r^2 θ
The triangle has area 1/2 r^2 sinθ
So the segment has area 1/2 r^2 (θ-sinθ)
To find the area of the segment bounded by the chord and the minor arc, you can follow these steps:
Step 1: Find the length of the chord.
Given that the radius of the circle is 12 cm and the angle at the center is 150°, we can apply the properties of a circle to find the length of the chord. The angle subtended by a chord at the center is twice the angle subtended by the same chord at any point on the circumference.
So, the angle subtended by the chord at any point on the circumference is 150°/2 = 75°.
Using the formula for the length of a chord in a circle, we can calculate it as follows:
Length of chord = 2 × radius × sin(angle/2)
Length of chord = 2 × 12 cm × sin(75°/2)
Step 2: Find the area of the sector.
The sector is the region bounded by the minor arc and the two radii originating from the center of the circle to the endpoints of the chord. The area of a sector can be calculated using the formula:
Area of sector = (angle/360°) × π × radius²
In this case, the angle of the sector is 150°, and the radius is 12 cm. So, the area of the sector would be:
Area of sector = (150°/360°) × π × (12 cm)²
Step 3: Find the area of the triangle formed by the chord and the two radii.
The triangle formed by the chord and the two radii is an isosceles triangle since the two radii are equal. The base of the triangle is the chord, and the height is the distance from the midpoint of the chord to the center of the circle.
To find the area of the triangle, we can use the formula:
Area of triangle = (1/2) × base × height
In this case, the base is the length of the chord found in step 1, and the height can be calculated using the formula:
Height = radius × cos(angle/2)
So, the area of the triangle is:
Area of triangle = (1/2) × (length of chord) × (height)
Step 4: Find the area of the segment.
The area of the segment is the difference between the area of the sector (step 2) and the area of the triangle (step 3).
Area of segment = Area of sector - Area of triangle
By performing these calculations using the given values, you should be able to find the area of the segment bounded by the chord and the minor arc of the circle.