Q. 134.6( 8 Votes )

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

Answer :

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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