Question

In Fig. 9, is shown a sector OAP of a circle with centre O, containing ∠ϴ. AB is perpendicular the radius OA and meets OP produced at B. Prove that the perimeter of the shaded region is
r [ tan theta + sec theta + π theta/180 - 1 ]

Answer 1

Perimeter of the shaded region

= arc AC + AB + BC

Arc AC = Φ * 2πr / 360

= Φ π r /180

Tan Φ = AB /OA

= AB /r

AB = r tan Φ

Similarly,

BC = OB -OC

= rsec Φ -r

= r(secΦ-1)

Perimeter = arc AC+AB+AC

= Φπr/180+ rtanΦ + rsecΦ-r

= r(tanΦ +sec Φ +π Φ/180-1)

☆Hence, your first part is proved !

Area :

= Area of right triangle OAB - Area of

sector

= 1/2 *OA *AB - Φ *πr^2 /360

= r^2tanΦ/2 - πr^2 Φ /360

= r^2 /2 ( tan Φ -π Φ/180)

Hence, proved!