Problem 257

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Problem 257: (2016 AMC 10a, 19)

In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$? $\textbf{(A) } 7 \qquad$ $ \textbf{(B) } 9 \qquad$ $ \textbf{(C) } 12 \qquad$ $ \textbf{(D) } 15 \qquad$ $ \textbf{(E) } 20$

Solution

Exercise 257: Solution

In rectangle $ABCD,$ $AB=12$ and $BC=8$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$?

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