A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle.
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Step-by-Step Solution
Step 1: Identify Given Values
We are given the radius of the circle, r=14 cm, and the angle subtended by the chord at the centre, θ=60∘. These are the key values needed for our calculations.
Step 2: Calculate Area of Sector
The area of a sector with angle θ and radius r is given by the formula 360∘θ×πr2. Substituting the given values, we get 360∘60∘×722×(14)2.
Step 3: Simplify Area of Sector
Simplifying the expression, 360∘60∘ becomes 61. Then, we multiply 61 by 722 and 196. This simplifies to 3308 cm2.
Step 4: Calculate Area of Triangle
The area of the triangle formed by the two radii and the chord can be calculated using the formula 21r2sinθ. Substituting the values, we get 21(14)2sin(60∘). We know that sin(60∘)=23.
Step 5: Simplify Area of Triangle
Simplifying the expression for the area of the triangle, we multiply 21 by 196 and then by 23. This results in 493 cm2.
Step 6: Calculate Area of Minor Segment
The area of the minor segment is found by subtracting the area of the triangle from the area of the sector. So, we subtract 493 from 3308.