1959 AHSME Problems/Problem 41

Problem

On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is: $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12$

Solution

$[asy] import geometry; point A = (0,4); point B = (-16,16); point C = (16,16); point D = (-16,0); point E = (16,0); point F = (0,16); // The line line l = line((-20,0),(20,0)); draw(l, Arrows); // Circles draw(circle(A,4)); dot(A); label("A",A,(0,-3)); draw(circle(B,16)); dot(B); label("B",B,W); draw(circle(C,16)); dot(C); label("C",C,(1,0)); //Tangency points dot(D); label("D",D,S); dot(E); label("E",E,S); dot(F); label("F",F,NE); // Triangle AFB, Segment BD draw(triangle(A,F,B)); draw(B--D); // Right angle labels markscalefactor=0.3; draw(rightanglemark(F,B,D)); draw(rightanglemark(B,D,E)); [/asy]$

$\fbox{D}$

1959 AHSC (Problems • Answer Key • Resources)
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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

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1959 AHSME Problems/Problem 41

Problem

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