QQuestion
O is the circumcentre of the triangle ABC. OD ⊥ BC. Prove that ∠BOD = ∠BAC.
✓Answer
Given: O is the circumcentre of triangle ABC. OD is perpendicular to BC.
To Prove: ∠BOD = ∠BAC
Proof:
O is the circumcentre, so it is equidistant from A, B, and C.
OD is perpendicular to BC, so it bisects the arc BC that does not contain A.
The angle subtended by the same arc at the centre is twice the angle subtended at the circumference.
Thus,
∠BOD = 2 ∠BCA
∠BAC = 2 ∠BCA
So, ∠BOD = ∠BAC (Hence proved)
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