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jmc

geometry junior

Problem

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?
problem
(A)
(B)
(C)
(D)
Solution
Let the radius of the large circle be . Then, the radius of the smaller circles are . The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is . This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is .
Final answer
D