2006 AMC 10A Problems/Problem 7

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Problem


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The $8\times18$ 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 $y$?

$\mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 10$

Solution

Since the two hexagons are going to be repositioned to form a square without overlap, the area will remain the same. The rectangle's area is $18\cdot8=144$. This means the square will have four sides of length 12. The only way to do this is shown below.

Square.JPG

As you can see from the diagram, the line segment denoted as $y$ is actually half as long as the side of the square, which leads one to conclude that its value is $\frac{12}{2} = 6 \Longrightarrow \mathrm{(A)}$.


See Also

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