MathematicsIf A, B, C are three points on a circle with center O such that ∠AOB= 90 °  and ∠BOC= 120 ° ,   then what will be the measure of ∠ABC  ?

If A, B, C are three points on a circle with center O such that AOB= 90 °  


and BOC= 120 ° ,   then what will be the measure of ABC  ?


  1. A
    60 °  
  2. B
    75 °  
  3. C
    90 °  
  4. D
    135 °   

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    Solution:

    It is given that A, B and C are three points on a circle with center O, AOB= 90 °   and BOC= 120 °  .
    Sum of measures of angles at a point is 360o.
    AOC+BOC+AOB= 360 o AOC= 360 o (BOC+AOB)  
    AOC= 360 o ( 120 ° + 90 o ) AOC= 150 o                  AOC   is at the center made by arc AC and ABC is subtended by arc AC on the circle.
    Angle subtended by the arc at the center is double the angle subtended by it at any point on the circle.
    ABC= 1 2 AOC ABC= 1 2 ×150 ABC= 75 o   Therefore, the correct option is (2).
     
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