MathematicsIn Fig. O is the center of the circle, ∠BCO = 30°. What is the value of  x and y.

In Fig. O is the center of the circle, ∠BCO = 30°. What is the value of  x and y.


  1. A
    x= 30 ° ,y= 15 °  
  2. B
    x= 30 ° ,y= 35 °  
  3. C
    x= 10 ° ,y= 15 °  
  4. D
    x= 15 ° ,y= 15 °   

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    Solution:

    Given that, BA, BD, BC are chords of the circle with center O and ∠BCO= 30 °  
    Joining AC and BO:
     Since we know the angle subtended by an arc at the center of circle will be double the angle subtended by it at any other point on the left part of the circle,
    So, 2∠ABD=∠AOD  .
    From given figure ∠AOD= 90 °  .
    So, 2∠ABD=∠AOD  .
    ⇒2∠ABD= 90 °   ⇒∠ABD= 45 °  
    Considering triangle OEC,
    Using the property sum of angles in a triangle is 180 °  .
    So, ∠COE+∠OEC+∠OCE= 180 °  .
     Using figure ∠OEC= 90 ° ,∠OCE= 30 °  .
    ∠COE+∠OEC+∠OCE= 180 °  
    ∠COE+ 90 ° + 30 ° = 180 °  
    ∠COE+ 120 ° = 180 °  
    ∠COE= 180 ° − 120 °   ∠COE= 60 °   As ∠DOC=∠DOE−∠COE  .
    ⇒∠DOC= 90 ° − 60 °   ⇒∠DOC= 30 °  
    Since the angle subtended by an arc at the center is two times the angle subtended by it at any point on the left part of the circle, so, ∠DBC=y= 1 2 ∠DOC  .
    ⇒y= 1 2 30 °   ⇒y= 15 °   Also, ∠ABE=y+ 45 °   ∠ABE= 15 ° + 45 °     ∠ABE= 60 °  
    Considering triangle ABE,
    We know that the sum of angles in a triangle is 180 °  .
    ∠BAE+∠ABE+∠AEB= 180 °   x+ 60 ° + 90 ° = 180 °   x= 180 ° − 150 °   x= 30 °   Therefore, option 1 is correct.
     
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