Problem 16

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Problem 16: (2016 AMC 10b, 19)

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?

$\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad$ $ \textbf{(B)}~\frac{\sqrt{2}}{13} \qquad$ $ \textbf{(C)}~\frac{9}{82} \qquad$ $ \textbf{(D)}~\frac{10}{91}\qquad$ $ \textbf{(E)}~\frac19$

Solution

Exercise 16: Solution

Rectangle $ABCD$ has $AB=6$ and $BC=5$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$, and point $F$ lies on $\overline{CD}$ so that $DF=1$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?

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