2014 AIME II Problem 3

Question

A rectangle has sides of length

a

and

36.

A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length

36.

The sides of length

a

can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length

a

parallel and separated by a distance of

24,

the hexagon has the same area as the original rectangle. Find

a^2.

Answer

Answer: 720

Concepts:Pythagorean Theorem trapezoid area

Difficulty rating: 2110

Small hint:

Each half of a

36

-side is a hinged bar of length

18

whose vertical extent in the hexagon is

12;

the Pythagorean theorem gives its horizontal reach

Big hint:

Cut the hexagon along the line through the two midpoint hinges into two congruent trapezoids and set the total area equal to

36a

Solution:

In the hexagon, each side of length

36

has folded at its midpoint into two bars of length

18.

The two sides of length

a

are

24

apart, so each bar spans a vertical distance of

12

and hence a horizontal distance of

\sqrt{18^2 - 12^2} = \sqrt{180} = 6\sqrt{5}.

The line through the two midpoint hinges splits the hexagon into two congruent trapezoids with parallel sides

a

and

a + 12\sqrt{5}

and height

12,

so the hexagon has area

\begin{aligned} &2 \cdot \frac{a + (a + 12\sqrt{5})}{2} \cdot 12 \\ &= 24a + 144\sqrt{5}. \end{aligned}

Setting this equal to the rectangle’s area

36a

gives

12a = 144\sqrt{5},

so

a = 12\sqrt{5}

and

a^2 = 720.