1986 AIME Problems/Problem 14
Problem
The shortest distances between an interior diagonal of a rectangular parallelepiped, , and the edges it does not meet are
,
, and
. Determine the volume of
.
Solution
![AIME 1986 Problem 14 sol.png](https://wiki-images.artofproblemsolving.com//e/e5/AIME_1986_Problem_14_sol.png)
In the above diagram, we focus on the line that appears closest and is parallel to . All the blue lines are perpendicular lines to
and their other points are on
, the main diagonal. The green lines are projections of the blue lines onto the bottom face; all of the green lines originate in the corner and reach out to
, and have the same lengths as their corresponding blue lines. So we want to find the shortest distance between
and that corner, which is
.
So we have:
Notice the familiar roots: ,
,
, which are
,
,
, respectively. (This would give us the guess that the sides are of the ratio 1:2:3, but let's provide the complete solution.)
We invert the above equations to get a system of linear equations in ,
, and
:
We see that ,
,
. Therefore
.
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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