2013 AMC 12B Problem 16

Problem 16 of 25IntermediateGeometry

Let

ABCDE

be an equiangular convex pentagon of perimeter

1.

The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let

s

be the perimeter of this star. What is the difference between the maximum and the minimum possible values of

s?

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Solution

An equiangular pentagon has all interior angles

108^\circ,

so each point of the star is an isosceles triangle with base angles

72^\circ

and apex

36^\circ.

By the equal base angles, each point contributes two sides that are the same fixed multiple

c

of the pentagon side it rests on. Summing over the five points, the star perimeter equals

2c\cdot(\text{pentagon perimeter}) = 2c,

independent of the individual side lengths. So

s

is constant, and the difference between its maximum and minimum values is

0.

Thus, the correct answer is A.