Yesterday, Integrated Learning posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
The length of any arc is the same fraction of its circumference as the arc angle is of 360°. We are told that the circle in this question has a radius of 10, so it must have a circumference of 20?. Arc ac has a length of 50?/3, which is 5/6 of 20?.
Thus, the arc is 5/6 of the circle, and has an arc angle of 300°. That means the opposite angle, the one opposite line segment ac is 60° (see figure).
Additionally, the radius of the circle is 10, and so each of the sides of the triangle coming out of the center are 10. This means that angle a and angle c must be equal, and since the angle in the center of the circle is 60°, they must also be 60°. This means the whole triangle is equilateral, and the third side, line segment ac, is also length 10.
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