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Figure shows a sector of a circle, centre O, containing an angle `theta`. Prove that (i) Perimeter of the shaded region is `r ( tan theta + sec theta + (pi theta)/180 -1)` (ii) Area of shaded region is `r^2/2 ( tan theta - pi theta/180)` |
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Answer» `tantheta=(AB)/r` `AB=rtantheta` `sectheta=(OB)/(OA)=(OB)/r` `OB=rsectheta` `PB=OB-OP=rsectheta-r` `arcAB=pirtheta/180^o` Perimeter=`rtantheta+rsectheta-r+(pirtheta)/180^0` `=r(tantheta+sectheta+pitheta/180^o-1)` Area of `/_OAB=1/2*r^2tantheta` Area of shaded region=`1/2r^2tantheta-pir^2theta/360` `=r^2/2(tantheta-pitheta/180^0)` |
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