a sector of OAP of a circle with centre O, containing angle theta. AB is perpendicular to the radius OA and meets OP produced at B. prove that the perimeter of shaded region is r[tan theta + sec theta + ¶theta/180 -1]
tan thetha = AB/OA
AB = OA tan thetha = r tan thetha
cos thetha = r/OB
OB = r/ cos thetha = r sec thetha
therefore BP = r sec thetha - r
PA = pie. r. thetha/180
therefore required perimeter = r tan thetha + pie. r. thetha/180 + r sec thetha -r = r( r tan thetha = pie. r. thetha/180 = sec thetha - 1)
proved.
AB = OA tan thetha = r tan thetha
cos thetha = r/OB
OB = r/ cos thetha = r sec thetha
therefore BP = r sec thetha - r
PA = pie. r. thetha/180
therefore required perimeter = r tan thetha + pie. r. thetha/180 + r sec thetha -r = r( r tan thetha = pie. r. thetha/180 = sec thetha - 1)
proved.