2008 AMC 12A Problems/Problem 18
Problem
A triangle with sides
,
,
is placed in the three-dimensional plane with one vertex on the positive
axis, one on the positive
axis, and one on the positive
axis. Let
be the origin. What is the volume if
?
Solution
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Without loss of generality, let be on the
axis,
be on the
axis, and
be on the
axis, and let
have respective lenghts of 5, 6, and 7. Let
denote the lengths of segments
respectively. Then by the Pythagorean Theorem,
so
; similarly,
and
. Since
,
, and
are mutually perpendicular, the tetrahedron's volume is
which is answer choice C.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
2008 AMC 12A (Problems • Answer Key • Resources) | |
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