A circular plot of land is divided into two regions by a chord of length 10√3 meters such that the chord subtends an angle of 120∘ at the center. Then, the area, in square meters, of the smaller region is
Question:
A circular plot of land is divided into two regions by a chord of length 10√3 meters such that the chord subtends an angle of 120∘ at the center. Then, the area, in square meters, of the smaller region is
A circular plot of land is divided into two regions by a chord of length 10√3 meters such that the chord subtends an angle of 120∘ at the center. Then, the area, in square meters, of the smaller region is
Options
Answer: 25(4π/3−√3)
Explanation:
Given: chord length = 10√3 m, central angle = 120°. Radius: chord = 2r sin(θ/2) = 2r sin60° = 2r(√3/2) = r√3 ⇒ r = (chord)/√3 = (10√3)/√3 = 10 Sector area (120°) = (120/360)·πr² = (1/3)·π·100 = 100π/3 Triangle area (isosceles with vertex angle 120°) = (1/2)r² sin120° = (1/2)·100·(√3/2) = 25√3 Smaller region (segment) = sector − triangle = 100π/3 − 25√3 = 25(4π/3 − √3) Answer: 25(4π/3 − √3)
Explanation:
Given: chord length = 10√3 m, central angle = 120°. Radius: chord = 2r sin(θ/2) = 2r sin60° = 2r(√3/2) = r√3 ⇒ r = (chord)/√3 = (10√3)/√3 = 10 Sector area (120°) = (120/360)·πr² = (1/3)·π·100 = 100π/3 Triangle area (isosceles with vertex angle 120°) = (1/2)r² sin120° = (1/2)·100·(√3/2) = 25√3 Smaller region (segment) = sector − triangle = 100π/3 − 25√3 = 25(4π/3 − √3) Answer: 25(4π/3 − √3)
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