Q13 of 22 Page 126

O is the circumcentre of Δ ABC and OD is perpendicular on the side BC; let us prove that ∠BOD = ∠BAC


By the theorem:-


The angle formed at the centre of a circle by an arc, is double of the angle formed by the same arc at any point on circle.


∠BOC = 2 ∠BAC ------------ (1)


In ΔBOD and ∠COD,


∠BDO = ∠CDO = 90° (given)


OD = OD (common)


OB = OC (radius)


Therefore, ΔBOD ≅ ∠COD by RHS congruency


⇒ ∠BOD = ∠COD ------------------- (2)


From (1) and (2), we get,


2 ∠BOD = 2 ∠BAC


⇒ ∠BOD = ∠BAC


Hence, proved.


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