2013 AMC 12B Problem 16

Question

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?

0

\dfrac{1}{2}

\dfrac{\sqrt5 - 1}{2}

\dfrac{\sqrt5 + 1}{2}

\sqrt5

Each interior angle of an equiangular pentagon is

108^\circ,

so every point of the star is an isosceles triangle with a

36^\circ

apex

Each star point’s two equal sides are the same fixed multiple

c

of the pentagon side it sits on, so the star perimeter is

2c

times the pentagon perimeter

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.

Problem 16 in other years