Q. 133.8( 6 Votes )

The circumcenter of the triangle ABC is O. Prove that OBC + BAC = 90°.

Answer :

Let ABC be the triangle whose circumcenter is O.


OBC = OCB = θ (opposite angles of equal sides


In ΔBOC, using the angle sum property of tringle, sum of all angles is 180°, we have:
BOC + OBC + OCB = 180°
BOC + θ + θ = 180°


BOC = 180° -2θ


Also, in a circle, angle subtended by an arc at the center is twice the angle subtended by it at any other point in the remaining part of the circle.
BOC = 2BAC


BAC = 1/2(BOC)


BAC = 1/2(180° - 2θ)


BAC = (90° - θ)


BAC + θ = 90°


BAC + OBC = 90°


Hence, proved.


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