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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10, SAT-II.
The inscribed angle ∠AMB=30° intercepts a circle of radius √2 at points A and B. What is the area of the shaded region?
The shaded region is a circle segment that is subtended by a central angle of 60°. Denoting the center of the circle with O, the triangle OAB must be equilateral.
To find the area of the shaded region we must subtract the area of the equilateral triangle OAB from the area of the circular sector OAB.
The sector is the sixth part of a circle, therefore its area is:
The side length of the triangle and the height are in the ratio:
Therefore, the height:
and the area: