MathematicsIf tangent at C  , intersects extended diameter  A B , of a circle at  D, if BD=BC   then ∠BAC  is equal to ____°.

If tangent at C  , intersects extended diameter  A B , of a circle at  D, if BD=BC   then BAC  is equal to ____°.


AB is a diameter of a circle and AC is its chord such that angleBAC=30^(@).  If the tangent at C intersects AB extended at D, then BC=BD.

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    Solution:

    If tangent at C  , intersects extended diameter  A B , of a circle at  D, if BD=BC   then BAC  is equal to 30°.
    AB is a diameter of a circle and AC is its chord such that angleBAC=30^(@).  If the tangent at C intersects AB extended at D, then BC=BD.Given: DC is the tangent. AB is the diameter. OB and OC are radius of circle.
    BD=BC   ….(Length of tangent drawn from external point is equal)
    OB = OC ….(Radius)
    Tangent is perpendicular to radius at point of contact.  OCD= 90 °   Angle subtended by semicircle is right angle.
    ACB= 90 °  
    In ΔOBC  ,  OCB=OBC   (angles of equal opposite sides)(1)
    In ΔBCD   ,
    BCD=CDB   ( angles of equal opposite sides) (2)
    By exterior angle theorem,
    OBC=BCD+CDB   OBC=2BCD  ….(From 2)
    OCD =90 OCB+BCD =90 OCB =90BCD  
    Put OBC   in place of OCB  ,
    OBC=90BCD  
    Put 2BCD   in place of OBC   ,
    2BCD =90BCD 3BCD =90 BCD =30  
    Put BCD=30  ,  OBC =2BCD OBC =2×30 OBC =60  
    In ACB  ,
    By angle sum property,
    ACB+CAB+ACB=180  
      90+CAB+60=180   CAB=30  
    Hence the measure of BAC   is equal to 30 °  .
     
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