2010 AMC 12A Problems/Problem 3

Problem 3

Rectangle $ABCD$ , pictured below, shares $50\%$ of its area with square $EFGH$ . Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$ . What is $\frac{AB}{AD}$ ?

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Solution

Let $EF = FG = GF = HE = s$ , let $AD = BC = h$ , and let $AB = CD = x$ .

$.2s^2 = hs$

$s = 5h$

$.5hx = hs$

$x = 2s = 10h$

$\frac{AB}{AD} = \frac{x}{h} = \boxed{10\ \textbf{(E)}}$

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2010 AMC 12A Problems/Problem 3

Problem 3

Solution