Difference between revisions of "1959 AHSME Problems/Problem 41"
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== Solution == | == 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> | ||
| + | |||
<math>\fbox{D}</math> | <math>\fbox{D}</math> | ||
Revision as of 19:21, 21 July 2024
Problem
On the same side of a straight line three circles are drawn as follows: a circle with a radius of 
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: 
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]](http://latex.artofproblemsolving.com/5/d/1/5d1b1f81b5a19339a249ba97d57945a7cb01ba05.png)
See also
| 1959 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 40 |
Followed by Problem 42 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
