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Question

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.

A
53
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B
37
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C
106
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D
212
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Solution

The correct option is C 106


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


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