2013 AMC 10A Problem 12

Problem 12 of 25IntermediateGeometry

\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?

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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

AD + DE + EF + AF

= AD + DB + FC + AF

= AB + AC

= 56.

Thus, C is the correct answer.

Tagged: similarity · parallelogram · perimeter