Question

Fill In The Blanks

If O is the circumcentre of ΔABC and D is the mid-point of the base BC, then ∠BOD = _______________.

Answer 1

Given:
O is the circumcentre of ∆ABC
D is the mid-point of the base BC

In ∆BOD and ∆COD,
OB = OC (radius)
BD = CD (D is the mid-point of the base BC)
OD = OD (common)

By SSS property,
∆BOD ≅ ∆COD

Therefore, ∠BOD = ∠COD (by C.P.C.T.)
⇒ ∠BOC = ∠BOD + ∠COD
⇒ ∠BOC = 2∠BOD ..(1)

We know, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle.
Thus, ∠BOC = 2∠BAC
⇒ 2∠BOD = 2∠BAC
⇒ ∠BOD = ∠BAC


Hence, ∠BOD = ∠BAC.