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Infinite sequence of circles

Source: 1971 AHSME Problem 35

April 24, 2014
ratiogeometryAMC

Problem Statement

Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is
<spanclass=′latex−bold′>(A)</span>(4+32):4<spanclass=′latex−bold′>(B)</span>92:2<spanclass=′latex−bold′>(C)</span>(16+122):1<span class='latex-bold'>(A) </span>(4+3\sqrt{2}):4\qquad<span class='latex-bold'>(B) </span>9\sqrt{2}:2\qquad<span class='latex-bold'>(C) </span>(16+12\sqrt{2}):1\qquad
<spanclass=′latex−bold′>(D)</span>(2+22):1<spanclass=′latex−bold′>(E)</span>3+22):1<span class='latex-bold'>(D) </span>(2+2\sqrt{2}):1\qquad <span class='latex-bold'>(E) </span>3+2\sqrt{2}):1
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