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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

If a point is placed randomly and with equal probability in the circle:

what is the probability that the Cartesian coordinates of the point satisfy:

(Calculator is allowed.)

We are working with a circle of radius 2 centered at (0, 2).

Write the absolute value explicitly to figure out what inequalities we have:

The graph of the inequality looks like two rays of slopes 1 and -1 meeting at the origin in a V-shape.

We compute the probability using the formula:

The useful area is the one that satisfies the two inequalities (is underneath the "V") and is also inside the circle. It is the shaded area in the figure above.

We can compute this area by dividing the circle in two parts: the part above the horizontal diameter where there are no useful points and the part under the horizontal diameter. In this useful part, only the points that are below the "V" are useful. Subtract from the area of the semicircle the area of the white triangle. This triangle is a 45-45-90 triangle because the two rays have slope 1 and -1 and therefore, form 45° angles with the axes. It can be divided into two smaller 45-45-90 triangles with legs of length 2. The area is:

The total area is the area of the circle (that is the area where the point may be placed).

The probability is: