Q15 of 22 Page 126

O is circumcentre of the triangle ABC and D is the mid point of the side BC. If ∠BAC = 40°, let us find the value of ∠BOD.

Since D is the midpoint of BC,


In ΔBOD and ΔCOD, we have,


BO = CO (radius)


OD = OD (common)


BD = DC (given D as midpoint)


∴ ΔBOD ≅ ΔCOD by SSS congruency


⇒ ∠BOD = ∠COD ------------- (1)


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


⇒ ∠BOD + ∠COD = 2 ∠BAC


From equation (1), we get,


⇒ 2 ∠BOD = 2 ∠BAC


⇒ ∠BOD = ∠BAC = 40°


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(i) The angle at the centre is ______ the angle on the circle, subtended by the same arc.


(ii) The length of two chord AB and CD are equal of a circle with centre O. If ∠APB and ∠DQC are angles on the circle, then the values of the two angles are_______.


(iii) If O is the circumcentre of equilateral triangle, then the value of the front angle formed by any side of the triangle is_______.

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