| Assertion (A): If the perimeter of a sector of a circle of radius \[\text{5}.\text{6 cm}\] is \[\text{27}.\text{2 cm},\] then the area of the sector is\[\text{44}.\text{8 c}{{\text{m}}^{\text{2}}}\]. |
| Reason (R): The area of a sector of a circle of radius r with central angle \[\theta \] is \[\frac{\theta }{360{}^\circ }\times \pi r\]. |
A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: C
Solution :
| [c] Radius of circle \[(r)=5.6m\] |
| Let \[\theta \] be the central angle of sector. |
| Perimeter of sector \[=r+r+\frac{\theta }{360{}^\circ }\times 2\pi r\] |
| \[\Rightarrow \,\,\,\,\,\,\,27.2=5.6+5.6+\frac{\theta }{360{}^\circ }\,\,\times 2\times \frac{22}{7}\times 5.6\] |
| \[\Rightarrow \,\,\,\,\,\,27.2=11.2+35.2\times \frac{\theta }{360{}^\circ }\Rightarrow 16=35.2\times \frac{\theta }{360{}^\circ }\]\[\Rightarrow \,\,\,\,\,\,\frac{\theta }{360{}^\circ }=\frac{16}{35.2}=\frac{16\times 10}{352}=\frac{5}{11}\] .(1) |
| Now, area of sector \[=\frac{\theta }{360{}^\circ }\times \pi {{r}^{2}}\] |
| \[=\frac{5}{11}\times \frac{22}{7}\times 5.6\times 5.6\] [using (i)] |
| \[=44.8c{{m}^{2}}\] |
| \[\therefore \] Assertion: True; Reason: False. |
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