Question

# Question 8 If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r , then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle is this statement false? Why?

Solution

## Let two circles C1 and C2 of radius r and 2r with centres O and O′ respectively. It  is given that, the arc length ˆAB of C1 is equal to arc of length ˆCD of C2 i.e. ˆAB=ˆCD=I (say). Now let θ1 be the angle subtended by arc ˆAB and θ2 be the angle subtended by arc ˆCD at the centre. ∴ˆAB=I=θ1360×2πr ˆCD=I=θ2360×2π(2r)=θ2360×4πr From Eqs. (i) and (ii) θ1360×2πr=θ2360×4πr ⇒θ1=2θ2 i.e angle formed by sector of C1 is double the angle formed by sector C2 at centre. Therefore, the given statement is true.

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