Mathematics
O is the circumcentre of the triangle ABC and D is mid-point of the base BC. Prove that ∠BOD = ∠A.
Circles
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Answer
From the below figure:
Arc BC subtends ∠BOC at center and ∠BAC at the point A on the circle.
∴ ∠BOC = 2∠A
In △OBD and △ODC,
OD = OD (Common side)
BD = CD (As D is the mid-point of BC)
OB = OC (Radius of the same circle)
∴ △OBD ≅ △ODC (SSS rule of congruency).
∴ ∠BOD = ∠COD (As corresponding part of congruent triangles are congruent.)
Since, ∠BOD = ∠COD so,
∠BOD = ∠BOC ….(i)
∠BOC = 2∠A
∠A = ∠BOC …..(ii)
From (i) and (ii) we get,
∠BOD = ∠A
Hence, proved that ∠BOD = ∠A.
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