MathematicsState whether the following statement is true or false.If O is the circumcenter of ΔABC  and OD is perpendicular on BC, we can say that ∠BOD=∠A  .

State whether the following statement is true or false.


If O is the circumcenter of ΔABC  and OD is perpendicular on BC, we can say that BOD=A  .


  1. A
    True
  2. B
    False 

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    Solution:

    It is given that O is the circumcenter of ΔABC and ODBC.
    Join points O and B.
    Arc BC subtends BOC at the center and BAC on the circle.
    By degree measure theorem,
     BOC=2  A                 ....(i)
    In right ΔOBD and ΔOCD,
    Side OD = OD                    …[Common]
    Hyp. OB =  hyp. OC           …[Radii of the circle]
    Therefore, ΔOBD=ΔOCD    …[RHS criterion]
    BOD=  COD=12  BOC
     BOC=2  BOD           ...(ii)
    From (i) and (ii),
    2  BOD=2  A
     BOD=  A
    Hence, the given statement is true.
    Therefore, the correct option is (1).
     
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