Difference between revisions of "1989 AIME Problems/Problem 12"
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Revision as of 18:17, 4 July 2013
Problem
Let be a tetrahedron with
,
,
,
,
, and
, as shown in the figure. Let
be the distance between the midpoints of edges
and
. Find
.
Solution
Call the midpoint of
and the midpoint of
.
is the median of triangle
. The formula for the length of a median is
, where
,
, and
are the side lengths of triangle, and
is the side that is bisected by median
. The formula is a direct result of the Law of Cosines applied twice with the angles formed by the median (Stewart's Theorem).
We first find , which is the median of
.
Now we must find , which is the median of
.
Now that we know the sides of , we proceed to find the length of
.
See also
1989 AIME (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 |
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