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In the Given Figure, O is the Centre of the Circle. Tangents at a and B Meet at C. If ∠Aco = 30°, Find: (I) ∠Bco (Ii) ∠Aob (Iii) ∠Apb - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

Question

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

Solution

In the given fig, O is the centre of the circle and CA and CB are the tangents to the circle from
C. Also, ∠ACO = 30°
P is any point on the circle. P and PB are joined.
To find: (i) BCO
(ii) `∠`AOB
(iii)`∠`APB
Proof:
(i) In ΔOAC and OBC
OC = OC (Common)
OA = OB (radius of the circle)
CA = CB (tangents to the circle)
∴  ΔOAC ≅  ΔOBC (SSS congruence criterion)
∴ `∠`ACO =  `∠`BCO = 30°
(ii)  ∴  `∠`ACB  = 30° + 30° = 60°
∴ `∠`AOB + `∠`ACB  = 180°
⇒ `∠`AOB  + 60° =  180°
⇒ `∠`AOB  = 180° -  60°
⇒ `∠`AOB  = 120°
(iii) Arc AB subtends `∠`AOB at the centre and `∠`APB is in the remaining part of the circle.
 ∴  `∠APB = 1/2  ∠AOB  =1/2 xx120 = 60°`

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Solution In the Given Figure, O is the Centre of the Circle. Tangents at a and B Meet at C. If ∠Aco = 30°, Find: (I) ∠Bco (Ii) ∠Aob (Iii) ∠Apb Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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