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

2013 AMC 12A Problem 9

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\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

Since

EF \parallel AB,

triangle

FEC

is similar to

ABC

and hence isosceles with

FE = FC

Half the perimeter equals

AF + FE = AF + FC = AC

Because

EF \parallel AB,

triangle

FEC

is similar to triangle

ABC,

which is isosceles, so

FE = FC.

Half the perimeter of parallelogram

ADEF

AF + FE

= AF + FC

= AC = 28.

The entire perimeter is

56.

Thus, the correct answer is C.