In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO
(ii) ∠AOB
(iii) ∠APB

Question

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find (i) ∠BCO (ii) ∠AOB (iii) ∠APB

KnowledgeBoat · Accepted Answer

(i) ∠BCO = ∠ACO = 30° (∵ C is the intersecting point of tangent AC and BC. So, OC divides ∠ACB in two halves.) Hence, the value of ∠BCO = 30°. (ii) We know that the tangent at any point of a circle and the radius through the point are perpendicular to each other. ∴ ∠OAC = ∠OBC = 90°. ∴ ∠AOC = ∠BOC (∵ tangents are equally inclined to the line joining the point and the centre of the circle.) Since sum of angles in a triangle = 180. In AOC ⇒ ∠AOC + ∠OAC + ∠ACO = 180° ⇒ ∠AOC + 90° + 30° = 180° ⇒ ∠AOC + 120° = 180° ⇒ ∠AOC = 180° - 120° = 60°. ∴ ∠BOC = 60°. From figure, ∠AOB = ∠AOC + ∠BOC = 60° + 60° = 120°. Hence, the value of ∠AOB = 120°. (iii) Arc AB subtends ∠AOB at the centre and ∠APB at the remaining part of the circle. ∴ ∠AOB = 2∠APB (∵ angle subtended at centre by an arc is double the angle subtended at remaining point of the circle.) ∠APB = x ∠AOB = x 120° = 60°. Hence, the value of ∠APB = 60°.

Mathematics

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO
(ii) ∠AOB
(iii) ∠APB

Circles

Related Questions

In the figure (i) given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z.

In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find
(i) ∠BOD
(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate
(i) ∠APT
(ii) ∠BOP

Mathematics

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find(i) ∠BCO(ii) ∠AOB(iii) ∠APB

Circles

Related Questions

In the figure (i) given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z.

In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find(i) ∠BOD(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate(i) ∠APT(ii) ∠BOP

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO
(ii) ∠AOB
(iii) ∠APB

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find
(i) ∠BOD
(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate
(i) ∠APT
(ii) ∠BOP