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Sum
Two arcs of the same lengths subtend angles of 60° and 75° at the centres of two circles. What is the ratio of radii of two circles?
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Solution
Let r1 and r2 be the radii of the two circles and let their arcs of same length S subtend angles of 60° and 75° at their centres.
Angle subtended at the centre of the first circle,
θ1 = 60°
= `(60 xx pi/180)^"c"`
= `(pi/3)^"c"`
∴ S = r1θ1 = `"r"_1(pi/3)` ...(i)
Angle subtended at the centre of the second circle,
θ2 = 75°
= `(75 xx pi/180)^"c"`
= `((5pi)/12)^"c"`
∴ S = r2θ2 = `"r"_2((5pi)/12)` ...(ii)
From (i) and (ii), we get
`"r"_1(pi/3) = "r"_2((5pi)/12)`
∴ `"r"_1/"r"_2 = 15/12`
∴ `"r"_1/"r"_2 = 5/4`
∴ r1 : r2 = 5 : 4.
Concept: Length of an Arc of a Circle
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