A circular garden of radius 10m has a straight line fence between the two points on the boundary of the garden. The fence lies inside the garden arena. This fence separates the walking area which is a small region from the plants. The fence is at a distance of 6 m from the centre of the garden. Find the angle subtended by the fence with the center of the garden. It is given that cos(53∘) = 35.
102 = OR2 + RQ2
102 = 62 +RQ2
⇒ RQ2 = 64
⇒ RQ = 8m
∴ PQ = 2RQ = 16m (OR is perpendicular bisector of PQ)
Area of ΔOPQ = 12 × Base × Height
= 12 × PQ × OR
= 12 × 16 × 6
=48m2
Area of sector OPSQ = ∠POQ360∘ × π×r2
In ORQ,
cos(∠ROQ) = adjacent sidehypotenuse
= OROQ
= 610
= 35
⇒ cos( ∠ROQ) = 35
It is given that cos(53∘) = 35.
Hence ∠ROQ = 53∘.
The angle subtended by the fence with the centre of the garden is ∠POQ
⟹∠POQ = 2(∠ROQ) = 106∘