Question

In Fig. 9, is shown a sector OAP of a circle with centre O, containing ∠θ. AB is perpendicular to the radius OQ and meets OP produced at B. Prove that the perimeter of shaded region is 

`r[tantheta+sectheta+(pitheta)/180-1]`

Answer 1

Perimeter of shaded region = AB + PB + arc length AP...(1)

Arc length AP = `theta/360xx2pir=(pithetar)/180" ....(2)"`

In right angled ΔOAB,

`tan theta=(AB)/r=>AB=rtan theta" ....(3)"`

`sec theta=(OB)/r =>OB=rsec theta`

OB = OP + PB

∴ r secθ=r+PB

∴ PB = r secθ - r.....(4)

Substitute (2), (3) and (4) in (1), we get

Perimeter of shaded region = AB+PB+ arc length AP

`=rtantheta+rsectheta-r+(pithetar)/180`

`=r[tantheta+sec theta+(pitheta)/180-1]`