In the figure, ∆ABC is inscribed in the circle. The triangle does not

Hi Scott ScottTargetTestPrep
Can you please explain how BC is major arc?

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When the vertex of a triangle expands from PR to AC and RQ to AB the vertex expands as a result of which the angle corresponding to it also expands as such it must be > 90

Among the given options only (E) fits in correctly !!

Solution:

If a triangle is inscribed in a circle such that it does not contain the center of the circle, it must be an obtuse triangle (note: if the center of the circle is on the inscribed triangle, it’s a right triangle, and if the center is inside the triangle, it’s an acute triangle). Since the triangle is obtuse, there must be an obtuse angle and the obtuse angle occurs at the vertex that contains partially the radius when it’s drawn from the vertex to the center of the circle. Notice that when the radius is drawn from B or C to center O, neither BO nor CO is contained in triangle ABC. However, when the radius is drawn from A to center O, AO is partially contained in triangle ABC. Therefore, angle A must be the obtuse angle and hence x > 90. Since 105 is the only number in the given choices that is greater than 90, then 105 is the correct answer.

Answer: E

A chord will divide the circle into two segments i.e major and minor segments.

The angle made by the chord/arc at the center will be twice the angle subtended by it at the circumference ( Major segment).
Also, the reflex angle made at the center will be twice the angle subtended by it at the circumference ( Minor segment).


If you have a clear overview of this property of circle, this question would be easy.

In the question here, the angle subtended by the chord CB at the minor segment is given as x.

By applying the above property, we can say that reflex angle COB = 2x.

Since reflex angle COB is greater than 180 °, we can conclude that x has to be greater than 90 °.

Analyzing the given answer choices, only answer option E satisfies this condition.

Option E is the right answer.

Thanks,

Clifin J Francis,
GMAT QUANT SME