Difference between revisions of "2003 AIME II Problems/Problem 7"
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| + | [[Category: Intermediate Geometry Problems]] | ||
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Revision as of 02:38, 6 December 2019
Problem
Find the area of rhombus 
given that the radii of the circles circumscribed around triangles 
and 
are 
and 
, respectively.
Solution
The diagonals of the rhombus perpendicularly bisect each other. Call half of diagonal BD 
and half of diagonal AC 
. The length of the four sides of the rhombus is 
.
The area of any triangle can be expressed as 
, where , , and 
are the sides and 
is the circumradius. Thus, the area of 
is 
. Also, the area of 
is 
. Setting these two expressions equal to each other and simplifying gives 
. Substitution yields 
and 
, so the area of the rhombus is 
.
See also
| 2003 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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. 
