Problem 411

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Problem 411: (2013 AMC 12a, 11)

Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?

$\textbf{(A) }1\qquad$ $ \textbf{(B) }\dfrac{3}{2}\qquad$ $ \textbf{(C) }\dfrac{21}{13}\qquad$ $ \textbf{(D) }\dfrac{13}{8}\qquad$ $ \textbf{(E) }\dfrac{5}{3}\qquad$

Solution

Exercise 411: Solution

Triangle $ABC$ is equilateral with $AB=12$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Triangle $ADE$ and trapezoid $DFGE$ have the same perimeter, and trapezoid $FBCG$ has perimeter $28$. What is $DE+FG$?

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