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

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 120°.

Explanation:

Let O be the centre and let the tangents at A and B meet at C.

Since OA ⟂ AC and OB ⟂ BC, in quadrilateral O A C B we have ∠OAC = ∠CB O = 90°.

The interior angles sum to 360°.

So 90° + ∠ACB + 90° + ∠AOB = 360°. 

Hence ∠ACB = 180° – ∠AOB

= 180° – 60° 

= 120°