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Question 7
O is the circumcentre of the ΔABC and D is the mid-point of the base BC. Prove that BOD=A.
 


Solution

Given in a ΔABC a circle is circumscribed having centre O.

Also, D is the mid point of BC.
To prove that BOD=A or BOD=BAC.
Construction : Join OB, OD and OC.

Proof
In ΔBOD and ΔCOD
OB = OC                                        [both are the radius of circle]
BD = DC                                        [D is the mid point of BC]
OD = OD                                     [Common side]
ΔBOD  ΔCOD        [ by SSS congruence rule]
BOD=COD     [by CPCT]   …..(i)

We know that in a circle, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle.
   2BAC=BOC
    BAC=12 BOC
    BAC=12 (BOD+COD)
    BAC=12×2(BOD) [ from Equation  (i)]
    BAC=BOD=A
Hence proved.

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