MathematicsIn the given figure , A,B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60° . If D is a point on the circle other than the arc ABC, find ∠ADC?

In the given figure , A,B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60° . If D is a point on the circle other than the arc ABC, find ∠ADC?


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  1. A
    45°
  2. B
    90°
  3. C
    120°
  4. D
    180° 

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    Solution:

    ∠BOC=30° and ∠AOB=60° (Given)
    ⇒∠AOC= ∠AOB + ∠BOC
    ⇒∠AOC= 60°+30°
    ⇒∠AOC= 90°
    We can see that AOC is the angle subtended by the arc ABC at its centre.
    D is now the point on the circle that is not the arc ABC.
    As a result, we know that the angle subtended by an arc at the centre is twice that of any other point on the circle.
    Therefore,
    ⇒ ∠AOC = 2×∠ADC
    ⇒ 90° = 2×∠ADC
    90°2=ADC
    ⇒ 45°= ∠ADC
    Hence we get the value of ∠ADC as 45°.
     
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