| The ratio of the area of a sector of a circle to the area of the circle is 1: 4. If the area of the circle is \[154\,\,c{{m}^{2}},\] the perimeter of the sector is [SSC (CGL) 2010] |
A) 20 cm
B) 25 cm
C) 36 cm
D) 40 cm
Correct Answer: B
Solution :
| \[\frac{\text{Area}\,\text{of}\,\text{sector}}{\text{Area}\,\text{of}\,\text{circle}}=\frac{1}{4}\] |
| \[\Rightarrow \] Area of sector \[=\frac{\text{Area}\,\text{of}\,\text{circle}}{\text{4}}\] |
| \[\Rightarrow \] \[\pi {{r}^{2}}\left( \frac{\theta {}^\circ }{360{}^\circ } \right)=\frac{\pi {{r}^{2}}}{4}\]\[\Rightarrow \]\[\theta {}^\circ =\frac{360{}^\circ }{4}=90{}^\circ \] |
| \[\because \] Area of circle \[=\pi {{r}^{2}}\] |
| \[\Rightarrow \] \[154=\pi {{r}^{2}}\] |
| \[\Rightarrow \] \[{{r}^{2}}=\frac{154\times 7}{22}\]\[\Rightarrow \]\[r=7\] |
| Perimeter of sector |
| \[=\frac{2\pi r\theta }{360{}^\circ }+2r=\frac{2\pi r\times 90{}^\circ }{360{}^\circ }+2r\] |
| \[=\frac{\pi r}{2}+2r=\frac{\pi \times 7}{2}+2\times 7\] |
| \[=11+14=25\,cm\] |
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