Difference between revisions of "2009 AIME II Problems/Problem 5"
(→Solution) |
|||
(One intermediate revision by the same user not shown) | |||
Line 39: | Line 39: | ||
<math>p =\frac {28}{20} = \frac {7}{5}</math>. The radius of circle <math>E</math> is <math>4 + \frac {7}{5} = \frac {27}{5}</math>, so the answer is <math>27 + 5 = \boxed{032}</math>. | <math>p =\frac {28}{20} = \frac {7}{5}</math>. The radius of circle <math>E</math> is <math>4 + \frac {7}{5} = \frac {27}{5}</math>, so the answer is <math>27 + 5 = \boxed{032}</math>. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=KKVxQV4hszo&t=7s | ||
== See Also == | == See Also == | ||
{{AIME box|year=2009|n=II|num-b=4|num-a=6}} | {{AIME box|year=2009|n=II|num-b=4|num-a=6}} | ||
+ | [[Category: Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:54, 9 August 2020
Contents
Problem 5
Equilateral triangle is inscribed in circle , which has radius . Circle with radius is internally tangent to circle at one vertex of . Circles and , both with radius , are internally tangent to circle at the other two vertices of . Circles , , and are all externally tangent to circle , which has radius , where and are relatively prime positive integers. Find .
Solution
Let be the intersection of the circles with centers and , and be the intersection of the circles with centers and . Since the radius of is , . Assume = . Then and are radii of circle and have length . , and angle degrees because we are given that triangle is equilateral. Using the Law of Cosines on triangle , we obtain
.
The and the terms cancel out:
. The radius of circle is , so the answer is .
Video Solution
https://www.youtube.com/watch?v=KKVxQV4hszo&t=7s
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.