(a) Find the area of the minor and the major sectors of a circle with radius 6 cm, if the angle subtended by the minor arc at the centre is 60∘. (Use π=3.14)
OR
(b) If a chord of a circle of radius 10 cm subtends an angle of 60∘ at the centre of the circle, find the area of the corresponding minor segment of the circle. (Use π=3.14 and 3=1.73)
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Step-by-Step Solution
Step 1: Identify Given Values
We are given the radius of the circle, r=10 cm, and the angle subtended by the chord at the center, θ=60∘. We need to find the area of the minor segment.
Step 2: Calculate Area of Sector
First, we calculate the area of the sector formed by the chord. The formula for the area of a sector is 360∘θ×πr2. Substituting the given values, we get 360∘60∘×3.14×(10 cm)2.
Step 3: Substitute Values and Compute Sector Area
Simplifying the fraction and performing the multiplication, the area of the sector is 61×314=52.33 cm2.
Step 4: Calculate Area of Triangle
Next, we calculate the area of the triangle formed by the two radii and the chord. Since the angle subtended at the center is 60∘ and the two sides are radii, the triangle is equilateral. The formula for the area of a triangle with two sides and the included angle is 21r2sinθ. Substituting the values, we get 21×(10 cm)2×sin(60∘).
Step 5: Substitute Values and Compute Triangle Area
We know that sin(60∘)=23. Using the given value 3=1.73, we calculate the area of the triangle as 21×100×21.73=50×0.865=43.25 cm2.
Step 6: Calculate Area of Minor Segment
Finally, the area of the minor segment is the difference between the area of the sector and the area of the triangle. Subtracting the triangle's area from the sector's area gives us the segment's area.
Step 7: Compute Final Segment Area
Subtracting the area of the triangle from the area of the sector, we get 52.33 cm2−43.25 cm2=9.08 cm2.