Difference between revisions of "1989 AIME Problems/Problem 12"
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− | Call the midpoint of <math>\overline{AB}</math> <math>M</math> and the midpoint of <math>\overline{CD}</math> <math>N</math>. <math>d</math> is the [[median]] of triangle <math>\triangle CDM</math>. The formula for the length of a median is <math>m=\sqrt{\frac{2a^2+2b^2-c^2}{4}}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of triangle, and <math>c</math> is the side that is bisected by median <math>m</math>. The formula is a direct result of the [[Law of Cosines]] applied twice with the angles formed by the median. | + | Call the midpoint of <math>\overline{AB}</math> <math>M</math> and the midpoint of <math>\overline{CD}</math> <math>N</math>. <math>d</math> is the [[median]] of triangle <math>\triangle CDM</math>. The formula for the length of a median is <math>m=\sqrt{\frac{2a^2+2b^2-c^2}{4}}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of triangle, and <math>c</math> is the side that is bisected by median <math>m</math>. 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 <math>CM</math>, which is the median of <math>\triangle CAB</math>. | We first find <math>CM</math>, which is the median of <math>\triangle CAB</math>. |
Revision as of 22:17, 26 January 2008
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 |