Question

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is ________.

Answer 1

AB is the chord of a circle with centre O such that ∠AOB = 60º. The tangents at A and B intersect at P.

Now,

∠OAP = 90º (Radius is perpendicular to the tangent at the point of contact)

Also, ∠OBP = 90º (Radius is perpendicular to the tangent at the point of contact)

In quadrilateral OAPB,

∠AOB + ∠OAP + ∠OBP + ∠APB = 360º (Angle sum property of quadrilateral)

⇒ 60º + 90º + 90º + ∠APB = 360º

⇒ ∠APB = 360º − 240º = 120º

Thus, the angle between the tangents at A and B is 120º.

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is __120º__.