2006 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2006 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 12A solutions, or check the answer key.

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Concepts:area square (geometry)

Difficulty rating: 1310

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 two hexagons form a square of area $8 \cdot 18 = 144,$ so the square has side $12.$

The staircase cut splits the width into three equal horizontal pieces of length $y,$ which together span the full width: $y + y + y = 18,$ so $y = 6.$ (The two vertical steps each rise $12 - 8 = 4,$ building the extra height of the square.)

Thus, the correct answer is A.

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