Problem 190

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Problem 190: (2009 AMC 10a, 21)

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

$\mathrm{(A)}\ 3-2\sqrt2 \qquad$ $ \mathrm{(B)}\ 2-\sqrt2 \qquad$ $ \mathrm{(C)}\ 4(3-2\sqrt2) \qquad$ $ \mathrm{(D)}\ \frac12(3-\sqrt2) \qquad$ $ \mathrm{(E)}\ 2\sqrt2-2$

Solution

Exercise 190: Solution

A window contains a ring of six congruent circles, each tangent to its two neighbors and internally tangent to a larger circle, as shown. What is the ratio of the area inside the larger circle that is not covered by the six smaller circles to the area of the larger circle?

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