2006 AMC 10A Problem 7

Problem 7 of 25EasierGeometry

The

8 \times 18

rectangle

ABCD

is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is

y?

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Solution

The rectangle’s area is

8 \cdot 18 = 144,

so the square formed has side

\sqrt{144} = 12.

Let

E

and

F

be the upper and lower endpoints of the stair-step cut. For the two congruent pieces to fit into a square, the three horizontal segments

DE,

the middle segment of length

y,

and

FB

must be equal. Together they span the rectangle’s width, so

DE+y+FB=18

gives

3y=18

and

y=6.

Indeed, the resulting square has side

2y=12,

agreeing with the area calculation. Thus, the correct answer is A.