2013 AMC 12A Problem 9

Problem 9 of 25EasierGeometry

\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

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.

Tagged: similarity · isosceles triangle · parallelogram