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Question
The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle of the sector in degrees, minutes and seconds
Solution
Let OAB be the sector of a circle of radius r.
The angle of the sector is θ.
Perimeter of the sector = OA + arc AB + OB arc AB = rθ
∴ Perimeter of the sector = r + r θ + r
= 2r + rθ
= r(2 + θ) ......(1)
Length of the arc of the semi – circle of radius
l = nπ ......(2)
Given that perimeter the circular sector = Length of the arc of the semi circle of radius r
From equations (1) and (2), we have
r(2 + θ) = πr
2 + θ = π
θ = π – 2
θ = `22/7 - 2 = (22 - 14)/7`
θ = `8/7` radians
θ = `(8/7 xx 180/pi)^circ`
= `8/7 xx 180/22 xx 7`
θ = `720/11`
= 65° 27' 16"
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