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.

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 and .

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.

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## Published by mathproblemsolvingskills

I coach students preparing for MathCounts and the AMCs, and I teach using curricula published by Art of Problem Solving.
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