The area of two semicircles are in the ratio 49:25. Find the ratio of their circumferences

Dear Student,

Please find below the solution to the asked query:

Let , Radius of both circles  =  r₁   and r₂   we know are of circle  = $\mathrm{\pi }$r²  , So from given information we get :

$$\begin{gathered} \frac{\mathrm{\pi }\mathit{ }{r_1}^2}{\mathrm{\pi } {r_2}^2} = \frac{49}{25} \\ \Rightarrow \frac{{\displaystyle {r_1}^2}}{{\displaystyle {r_2}^2}} = \frac{{\displaystyle 7^2}}{{\displaystyle 5^2}} \\ \Rightarrow \frac{{\displaystyle r_1}}{{\displaystyle r_2}} = \frac{{\displaystyle 7}}{{\displaystyle 5}} \\ \mathbf{\Rightarrow }{\mathit{r}}_{\mathbf{1}}\mathbf{ }\mathbf{=}\mathbf{ }\frac{{\displaystyle 7}}{{\displaystyle 5}}{\mathit{r}}_{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{\hspace{0.17em}}\mathbf{)} \end{gathered}$$

We know circumference of circle  = 2 $\mathrm{\pi }$r

Then,
$$\begin{gathered} \frac{\mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{one} }{\mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{two}} = \frac{2{\mathrm{\pi r}}_1}{2{\mathrm{\pi r}}_2} \\ \Rightarrow \frac{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{one} }}{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{two}}} = \frac{{\displaystyle {\mathrm{r}}_1}}{{\displaystyle {\mathrm{r}}_2}} \\ \Rightarrow \frac{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{one} }}{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{two}}} = \frac{{\displaystyle \frac{7}{5}{\mathrm{r}}_2}}{{\displaystyle {\mathrm{r}}_2}} ( \mathrm{From} \mathrm{equation} 1 ) \\ \mathbf{\Rightarrow }\frac{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{one} }}{{\displaystyle \mathrm{Circumference} \mathrm{of} \mathrm{circle} \mathrm{two}}}\mathbf{ }\mathbf{=}\mathbf{ }\frac{{\displaystyle 7}}{{\displaystyle 5}} \end{gathered}$$
Therefore,

Ratio of their circumferences = 7 : 5                                             ( Ans )


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