Question

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 $y?$

$6$

$7$

$8$

$9$

$10$

Answer: A

Concepts:area area decomposition

Difficulty rating: 1190

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.

2006 AMC 10A Problem 7

Answer: A

Difficulty rating: 1190

Solution:

Problem 7 in Other Years