2006 AMC 10A Problem 7

Below is the professionally curated solution for Problem 7 of the 2006 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 10A solutions, or check the answer key.

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Concepts:area area decomposition

Difficulty rating: 1190

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 $y?$

$6$

$7$

$8$

$9$

$10$

Solution:

The rectangle's area is $8 \cdot 18 = 144,$ so the square formed has side $\sqrt{144} = 12.$

Along the top edge the three equal horizontal pieces satisfy $DE = y = FB$ with $DE + y + FB = 18.$ Hence $3y = 18,$ so $y = 6.$

Thus, the correct answer is A.

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