2001 AIME I Problems/Problem 12
Problem
A sphere is inscribed in the tetrahedron whose vertices are and
The radius of the sphere is
where
and
are relatively prime positive integers. Find
Solution
import three; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(5,-10,4); pair A=(6,0,0), B=(0,4,0), C=(0,0,2), D=(0,0,0); D(MP("A",A)--MP("B",B)--MP("C",C,N)--A--MP("D",D)--B--D--C); (Error making remote request. Unknown error_msg)
Connect all four vertices of tetrahedron to its incenter,
. This yields four tetrahedra
, all of which have height of
(the radius of the inscribed sphere), and which together form
. It follows that
where is the surface area of
.
Since all lie on the planes containing the axes, their areas are straightforward to calculate; respectively
. To find
, we can using the 3-dimensional distance formula (
) to find that
. From here, we can use the Law of Cosines and the sine area formula to compute
, or we can use a manipulated version of Heron's formula:
.[1]
Thus, . The volume of
we can compute by letting
to be the height to face
, so
. Therefore,
, and
.
![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
See also
- <url>viewtopic.php?p=384205#384205 Discussion on AoPS</url>
2001 AIME I (Problems • Answer Key • Resources) | ||
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