2013 AMC 10A Problem 12

Question

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

Use the parallels to get similar triangles near

B

and

C

Rewrite opposite sides of the parallelogram as pieces of

AB

and

AC

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.

Problem 12 in other years