The given figure shows a circle with center O. Points A, D, B and C lie on the circumference of the given circle and angle AOB = 108°. If arc ADB = 2 × arc BC, find :
(i) ∠ACB
(ii) ∠CAB
(iii) ∠ADB

Question

The given figure shows a circle with center O. Points A, D, B and C lie on the circumference of the given circle and angle AOB = 108°. If arc ADB = 2 × arc BC, find :
(i) ∠ACB
(ii) ∠CAB
(iii) ∠ADB

KnowledgeBoat · Accepted Answer

(i) Join BC. Given, ⇒ arc ADB = 2 × arc BC ⇒ ∠AOB = 2∠BOC ∠BOC = = 54°. In △ AOC, ⇒ OA = OC (Radius of circle) ⇒ ∠OAC = ∠OCA = y (let) [Angles opposite to equal sides are equal] By angle sum property of triangle, ⇒ ∠OAC + ∠OCA + ∠AOC = 180° ⇒ y + y + (108° + 54°) = 180° [∵ ∠AOC = ∠AOB + ∠BOC] ⇒ 2y = 180° - 162° ⇒ 2y = 18° ⇒ y = = 9°. ⇒ ∠OCA = 9° In △ BOC, ⇒ OB = OC (Radius of circle) ⇒ ∠OBC = ∠OCB = x (let) [Angles opposite to equal sides are equal] By angle sum property of triangle, ⇒ ∠OBC + ∠OCB + ∠BOC = 180° ⇒ x + x + 54° = 180° ⇒ 2x = 180° - 54° ⇒ 2x = 126° ⇒ x = = 63°. ⇒ ∠OCB = 63° From figure, ⇒ ∠ACB = ∠OCB - ∠OCA = 63° - 9° = 54°. Hence, ∠ACB = 54°. (ii) We know that, Angle which and arc subtends at the center is double that which it subtends at any point on the remaining part of the circumference of the circle. ∴ ∠BOC = 2∠CAB ⇒ ∠CAB = = 27°. Hence, ∠CAB = 27°. (iii) Arc AB subtends ∠AOB at the centre and ∠ACB at the remaining part of the circle. ∴ ∠AOB = 2∠ACB ∠ACB = = 54°. In cyclic quadrilateral ADBC, ⇒ ∠ADB + ∠ACB = 180° [∵ Sum of opp ∠'s in cyclic quadrilateral = 180°] ⇒ ∠ADB + 54° = 180° ⇒ ∠ADB = 180° - 54° = 126°. Hence, ∠ADB = 126°.

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Mathematics

The given figure shows a circle with center O. Points A, D, B and C lie on the circumference of the given circle and angle AOB = 108°. If arc ADB = 2 × arc BC, find :
(i) ∠ACB
(ii) ∠CAB
(iii) ∠ADB

Circles

Related Questions

If a, c, a² + b² and b² + c² are in proportion, show that b is mean proportion between a and c.

Mathematics

The given figure shows a circle with center O. Points A, D, B and C lie on the circumference of the given circle and angle AOB = 108°. If arc ADB = 2 × arc BC, find :(i) ∠ACB(ii) ∠CAB(iii) ∠ADB

Circles

Related Questions

If a, c, a2 + b2 and b2 + c2 are in proportion, show that b is mean proportion between a and c.

The given figure shows a circle with center O. Points A, D, B and C lie on the circumference of the given circle and angle AOB = 108°. If arc ADB = 2 × arc BC, find :
(i) ∠ACB
(ii) ∠CAB
(iii) ∠ADB

If a, c, a² + b² and b² + c² are in proportion, show that b is mean proportion between a and c.