Question

# Question 7 O is the circumcentre of the ΔABC and D is the mid-point of the base BC. Prove that ∠BOD=∠A.

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Solution

## Given in a ΔABC a circle is circumscribed having centre O. Also, D is the mid point of BC. To prove that ∠BOD=∠A or ∠BOD=∠BAC. Construction : Join OB, OD and OC. Proof In ΔBOD and ΔCOD OB = OC [both are the radius of circle] BD = DC [D is the mid point of BC] OD = OD [Common side] ΔBOD ≅ ΔCOD [ by SSS congruence rule] ∠BOD=∠COD [by CPCT] …..(i) We know that in a circle, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle. ∴ 2∠BAC=∠BOC ⇒ ∠BAC=12 ∠BOC ⇒ ∠BAC=12 (∠BOD+∠COD) ⇒ ∠BAC=12×2(∠BOD) [ from Equation (i)] ⇒ ∠BAC=∠BOD=∠A Hence proved.

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