# 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? - Mathematics and Statistics

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?

#### 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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