Why so big?

Geometry Level 4

The six numbers in each set below—not necessarily in that order—are possible side lengths of a hexagon:

Set A: {14, 188, 198, 237, 372, 421}
Set B: {17, 92, 189, 256, 331, 428}.

If the hexagon in question has all congruent interior angles, which set has valid side lengths of the hexagon?

What set of side lengths is possible?

Set A Set B Both sets Neither set

1 solution

Jeremy Galvagni
Aug 2, 2018

In an equiangular hexagon, for any pair of parallel sides, the sum of the sides on one side equals the sum of sides on the other. In other words, if the sides in order are A,B,C,D,E,F then $A+B=D+E$ , $B+C=E+F$ , and $C+D=F+A$ .

So what we can do is find the sum of all possible pairs and see if there are any matches. Six numbers have 15 pairs. I will not list them all, but for set A the equal pairs are:

$14+372=188+198$ , $372+237=188+412$ , and $237+198=421+14$

And for set B:

$17+428=256+189$ , $428+92=189+331$ , and $92+256=331+17$

The fact that both sets have numbers that work this way means that $\boxed{\text {both sets}}$ have valid side lengths for an equiangular hexagon.

At the start, you probably meant to say "equiangular" rather than "equilateral."

Marta Reece - 2 years, 10 months ago

Of course. Thank you. Nice problem.

Jeremy Galvagni - 2 years, 10 months ago

Why so big?

1 solution

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