Difference between revisions of "2000 AIME II Problems/Problem 12"
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{{AIME box|year=2000|n=II|num-b=11|num-a=13}} | {{AIME box|year=2000|n=II|num-b=11|num-a=13}} |
Revision as of 19:35, 18 March 2008
Problem
The points ,
and
lie on the surface of a sphere with center
and radius
. It is given that
,
,
, and that the distance from
to triangle
is
, where
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Solution
Let be the foot of the perpendicular from
to the plane of
. By the Pythagorean Theorem on triangles
,
and
we get:
It follows that , so
is the circumcenter of
.
By Heron's Formula the area of is:
Now the circumradius of is:
Thus by the Pythagorean Theorem again,
.
So the final answer is
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |