Difference between revisions of "2014 AMC 10B Problems/Problem 22"
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==Problem== | ==Problem== | ||
| + | Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles? | ||
| + | |||
| + | <math>\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3</math> | ||
| + | |||
| + | <asy> | ||
| + | scale(200); | ||
| + | draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); | ||
| + | path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); | ||
| + | draw(p); | ||
| + | p=rotate(90)*p; draw(p); | ||
| + | p=rotate(90)*p; draw(p); | ||
| + | p=rotate(90)*p; draw(p); | ||
| + | draw(scale((sqrt(5)-1)/4)*unitcircle); | ||
| + | </asy> | ||
==Solution== | ==Solution== | ||
Revision as of 19:14, 20 February 2014
Problem
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
![[asy] scale(200); draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); draw(scale((sqrt(5)-1)/4)*unitcircle); [/asy]](http://latex.artofproblemsolving.com/a/a/3/aa3449d7a1f5549a72c1e5be22ee89bd4499ee51.png)
Solution
See Also
| 2014 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
