We are given a tetrahedron with two edges of length a and the remaining four edges of length b where a and b are positive real numbers. What is the range of possible values for the ratio v=a/b?
The two equal edges would be skew lines at right angles in projection where they would look like they intersect in their centers. If they were almost touching (in the limit) a/b would be sqrt(2). As the two equal edges get farther and farther apart a/b can will approach zero as b can be as high as you like.
However, if the two that are equal to a are adjacent to each other the ratio can go from 1 to zero, exclusive. The 1 ratio would be degenerate as a straight line as the angle between the two size-a edges became 180°. The zero limit would be if the two size-a edges approached an agle of zero and the other four sides could be as large as you want.
Overall, then, the range is zero to sqrt(2), exclusive, being an open interval.
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Posted by Charlie
on 2024-06-05 20:23:55 |
proposed solution
