Given figure shows sector OAB and radius 54 cm
Another circle XYZ with centre P, is enclosed by the sector OAB. If angle AOB= 60 Find the area of OXPY.

Question

Given figure shows sector OAB and radius 54 cm
Another circle XYZ with centre P, is enclosed by the sector OAB If
angle AOB= 60 Find the area of OXPY

Manbar Singh · Accepted Answer

Join OP
In ∆OPX and ∆OPYOX = OY  Tangents drawn from an external point to a circle are equal in lengthOP = OP   CommonOX = OY  Radii of same circleSo, ∆OPX ≅ ∆OPY  SSS⇒∠POX = ∠POY  CPCTSo, ∠POX = ∠POY =12∠AOB = 12×60° = 30°Now, OAB is a sector of the circle, thenOA = OZ = OB = 54 cmNow, in ∆OYP,sin 30° = PYOP⇒12 = r54 - r⇒2r = 54 - r⇒r = 18 cmNow, cos 30° = OYOP⇒32 = OY54 - r⇒32 = OY54 - 18⇒OY = 183 cmNow, area of ∆OYP = 12×OY×PY=12×183×18=1623 cm2Similarly, area of ∆OXP = 1623 cm2So, area of OXPY = area of ∆OYP +area of ∆OXP=1623 + 1623=3243 cm2

Given figure shows sector OAB and radius 54 cm Another circle XYZ with centre P, is enclosed by the sector OAB. If angle AOB= 60 Find the area of OXPY.

Given figure shows sector OAB and radius 54 cm
Another circle XYZ with centre P, is enclosed by the sector OAB. If angle AOB= 60 Find the area of OXPY.