# Trapezoid existence

Not all 3 line segments will form a triangle.  If you don’t believe me, try forming a triangle using line segments of lengths 2, 2 and 20.  The two sides of length 2 aren’t long enough to span the 3rd side.  A quick test for this is to see that each side is smaller than the combined lengths of the other 2 sides.  And that each side is greater than the difference between the other 2 sides.

This week I learned that there is also a fun test to determine whether 4 given side lengths, when assigned to be either the parallel or non-parallel sides can actually form a trapezoid.  Not all 4 side lengths in a particular orientation can form a trapezoid.  The formula seems weird, but it’s actually an extension of the triangle inequality.  Here a and b are the lengths of the parallel sides and c and d are the non-parallel sides.

$|d-c|<|b-a|

To form the triangle, draw a line from a vertex parallel to a non-parallel side to the opposite parallel side (see below).  The triangle has side lengths $b-a, c,$ and $d$.

An old AMC gives 4 side lengths of possible trapezoids, and the student must test each orientation to determine whether a trapezoid can be formed.