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The Figure Shows Two Circles Which Intersect at a and B. the Centre of the Smaller Circle is O and Lies on the Circumference of the Larger Circle. Given ∠Apb = A°. Calculate, in Terms of A°, the - 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

The figure shows two circles which intersect at A and B. The centre of the smaller circle is O
and lies on the circumference of the larger circle. Given ∠APB = a°.
Calculate, in terms of a°, the value of:
(i) obtuse ∠AOB,
(ii) ∠ACB
(iii) ∠ADB.
Give reasons for your answers clearly.

Solution

 

(i) obtuse ∠AOB = 2°∠APB = 2a° 
(Angle at the centre is double the angle at the circumference subtended by the same chord)

(ii)  OABC is a cyclic quadrilateral
∠AOB + ∠ACB = 180° 
(Pair of opposite angles in a cyclic quadrilateral are supplementary)
 ⇒ ∠ACB = 180° - 2a° 

(iii) Join AB.
∠ADB = ∠ACB =180° - 2a° 
(Angle subtended by the same arc on the circle are equal)

  Is there an error in this question or solution?
Solution The Figure Shows Two Circles Which Intersect at a and B. the Centre of the Smaller Circle is O and Lies on the Circumference of the Larger Circle. Given ∠Apb = A°. Calculate, in Terms of A°, the 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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