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In the Figure, O is the Centre of the Circle and the Length of Arc Ab is Twice the Length of Arc Bc. If Angle Aob = 108°, Find: - ICSE Class 10 - Mathematics

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ConceptArc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle

Question

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108°, find:

(i) ∠CAB  (ii) ∠ADB

Solution


(i) Join AD and DB
Arc B = 2 arc BC and ∠AOB = 180°
∴ ∠BOC = 1 ∠AOB
              = `1/2 xx108°`
              = 54°
Now, Arc BC subtends ∠BOC at the centre and
∠CAB at the remaining part of the circle.
∴ `∠CAB = 1/2 ∠BOC`
               =` 1/2xx  54°`
               = 27°

(ii) Again, Arc AB subtends ∠AOB at the centre and
   ∠ACB at the remaining part of the circle.
`∠ ACB = 1/2 ∠AOB`
            =` 1/2 xx 108°`
            = 54
In cyclic quadrilateral ADBC
∠ADB + ∠ACB =180°        [sum of opposite angles]
⇒ ∠ADB + 54° = 180°
⇒ ∠ADB = 180° - 54°
⇒ ∠ADB = 126°

  Is there an error in this question or solution?
Solution In the Figure, O is the Centre of the Circle and the Length of Arc Ab is Twice the Length of Arc Bc. If Angle Aob = 108°, Find: Concept: Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle.
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