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
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 | |
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