True or False
If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.
This statement is False.
Consider the given figure. In which we have a circle with centre O and AB a chord with ∠AOB = 60°
Since, tangent to any point on the circle is perpendicular to the radius through point of contact,
OA ⏊ AC and OB ⏊ CB
∠OBC = ∠OAC = 90° .......[Equation (1)]
Using angle sum property of quadrilateral in Quadrilateral AOBC,
∠OBC + ∠OAC + ∠AOB + ∠ACB = 360°
90° + 90° + 60° + ∠ACB = 360°
∠ACB = 120°
Hence, the angle between two tangents is 120°.
Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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