Difference between revisions of "2004 AMC 10B Problems/Problem 16"
(New page: == Problem == Three circles of radius <math>1</math> are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? <math> \mathr...) |
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Three circles of radius <math>1</math> are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? | Three circles of radius <math>1</math> are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? | ||
| − | <math> \mathrm{(A) \ } \frac{2 + \sqrt{6}}{3} \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } \frac{2 + 3\sqrt{2}}{2} \qquad \mathrm{(D) \ } \frac{3 + 2\sqrt{ | + | <math> \mathrm{(A) \ } \frac{2 + \sqrt{6}}{3} \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } \frac{2 + 3\sqrt{2}}{2} \qquad \mathrm{(D) \ } \frac{3 + 2\sqrt{3}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} </math> |
==Solution== | ==Solution== | ||
Revision as of 10:16, 25 March 2011
Problem
Three circles of radius 
are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
Solution
The situation in shown in the picture below. The radius we seek is 
. Clearly 
. The point 
is clearly the center of the equilateral triangle 
, thus 
is 
of the altitude of this triangle. We get that 
. Therefore the radius we seek is
.
WARNING. Note that the answer does not correspond to any of the five options. Most probably there is a typo in option D.
![[asy] unitsize(2cm); pair A=(0,0), B=dir(0)*2, C=dir(60)*2; draw(circle(A,1)); draw(circle(B,1)); draw(circle(C,1)); dot(A); dot(B); dot(C); draw(A--B--C--cycle); pair D=A+dir(210), E=B+dir(-30), F=C+dir(90); draw(circumcircle(D,E,F)); dot(D); dot(E); dot(F); pair S=(A+B+C)/3; dot(S); draw(S--D); draw(S--E); draw(S--F); label("$S$",S,S); label("$A$",A,SE); label("$D$",D,SW); label("$B$",B,NE); label("$C$",C,ENE); [/asy]](http://latex.artofproblemsolving.com/0/5/6/056b92c537632c2d904effcbd77cab0e9d61be72.png)
See also
| 2004 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
