2013 AMC 10A Problem 12

Below is the professionally curated solution for Problem 12 of the 2013 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 10A solutions, or check the answer key.

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Difficulty rating: 1420

12.

In $\triangle ABC,$ $AB=AC=28$ and $BC=20.$ Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF?$

$48$

$52$

$56$

$60$

$72$

Solution:

Note that $\triangle DBE \sim \triangle ABC$ and $\triangle FEC \sim \triangle ABC$ due to the parallel lines.

This tells us that $DB = DE$ and $FE = FC.$ We have that the perimeter of $ADEF$ is $AD + DE + EF + AF$ $= AD + DB + FC + AF$ $= AB + AC$ $= 56.$

Thus, C is the correct answer.

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