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  .

# State whether the following statement is true or false.If O is the circumcenter of  and OD is perpendicular on BC, we can say that  .

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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