Difference between revisions of "2011 AMC 12A Problems/Problem 17"
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dotfactor=4; | dotfactor=4; | ||
− | pair A=(0,0), B=( | + | pair A=(0,0), B=(3,0), C=(0,4); |
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dot (A); | dot (A); | ||
dot (B); | dot (B); | ||
dot (C); | dot (C); | ||
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draw(Circle(A,1)); | draw(Circle(A,1)); | ||
− | draw(Circle(B, | + | draw(Circle(B,2)); |
− | draw(Circle(C, | + | draw(Circle(C,3)); |
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</asy> | </asy> | ||
Revision as of 16:18, 22 September 2013
Problem
Circles with radii ,
, and
are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
Solution
The centers of these circles form a 3-4-5 triangle, which has an area equal to 6.
The 3 triangles determined by one center and the two points of tangency that particular circle has with the other two are, by Law of Sines,
which add up to . Thus the area we're looking for is
.
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.