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If in two circles arcs of the same length subtend angles $60^{\large\circ}$ and $75^{\large\circ}$ at the centre,find the ratio of their radii

$\begin{array}{1 1}(A)\;5 : 4&(B)\;4 : 5\\(C)\;5 : 3&(D)\;6 : 5\end{array} $

1 Answer

  • $\theta=\large\frac{l}{r}$
  • $\theta$=angle subtended by arc
  • $l$=length of the arc
  • $r$=radius of the circle
  • $1^{\large\circ}=(\large\frac{\pi}{180})$ radian=0.01746(approx)
Let the length of the circle be $l$
Angle of the circle 1 =$60^{\large\circ}$
Angle of the circle 2 =$75^{\large\circ}$
Let the radius be $r_1$ and $r_2$
$\Rightarrow r_1\times 60\times \large\frac{\pi}{180}$
$\Rightarrow \large\frac{\pi r_1}{3}$
$\Rightarrow r_2\times 75\times \large\frac{\pi}{180}$
$\Rightarrow \large\frac{5\pi r_2}{12}$
Since $l$ is same for both the circles
$\Rightarrow \large\frac{\pi r_1}{3}=\frac{5\pi r_2}{12}$
$\Rightarrow r_1 : r_2=5 : 4$
Hence (A) is the correct answer.
answered Apr 16, 2014 by sreemathi.v

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