MathematicsIn the given figure, O   is the centre of the circle. Determine ∠APC ,if DA and DC are tangents and ∠ADC=50∘.

In the given figure, O   is the centre of the circle. Determine ∠APC ,if DA and DC are tangents and ∠ADC=50∘.


  1. A
    105∘
  2. B
    100∘
  3. C
    115∘
  4. D
    120∘ 

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    Solution:

    Given that DA and DB are the tangents to the circle with centre O.  ∠ADC=50∘.
    To find ∠APC, join AO and CO.
    The radius and tangent are perpendicular at their point of contact.
    Use the property that the radius and tangent are perpendicular at their point of contact so find ∠DAO and ∠DCO.
    ∠DAO= 90 ° ∠DCO= 90 °  
    Now consider the quadrilateral AOCD and use the property that sum of all angles of a quadrilateral is 360∘.
    ∠ADC+∠DAO+∠AOC+∠OCD=360∘
    ⇒50∘+90∘+∠AOC+90∘=360∘
    ⇒∠AOC+230∘=360∘
    ⇒∠AOC=130∘          ………..(1)
    Use the central angle theorem that the central angle from two chosen points A and C on the circle is always twice the inscribed angle from those two points.
    ∠APC=360∘-130∘2   ⇒∠APC=115∘ Hence, ∠APC=115∘.
    Therefore, option 3 is correct.
     
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