Quant Question Of The Day: 34
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Geometry
In triangle ABC, ∠BAC = 60° and O is a point inside the triangle such that ∠BOC = 120°. Which of the following is certainly true?
(1) O is orthocenter of the triangle ABC.
(2) O is circumcenter of the triangle ABC.
(3) Both (1) and (2) are true.
(4) None of (1) and (2) is certain.
Let P be the circumcenter of the triangle, then certainly BPC = 120. But if we draw a circle passing through B, P, C, and mark a point O on the arc B-P-C other than P, then BOC will also be 120. Same is true for orthocenter also.
Thus O need not to be orthocenter or circumcenter of the triangle ABC.
Option 3
When the figure is drawn it is evident that the vertices lie on the sides of circle because angle at the center is double made at the circumference Making BO and OC radius which makes all angles 60degrees of triangle. Hence for equilateral triangle circumcenter, orthocenter is the same point.
3) both 1 and 2 are true
option 3