2018 AMC 8

Complete problem set with solutions and individual problem pages

Problem 20 Hard

In \triangle ABC, a point E is on \overline{AB} with AE=1 and EB=2. Point D is on \overline{AC} so that \overline{DE} \parallel \overline{BC} and point F is on \overline{BC} so that \overline{EF} \parallel \overline{AC}. What is the ratio of the area of CDEF to the area of \triangle ABC?

  • A.

    \frac49

  • B.

    \frac12

  • C.

    \frac59

  • D.

    \frac35

  • E.

    \frac23

Answer:A

By similar triangles, we have [ADE] = \frac{1}{9}[ABC]. Similarly, we see that [BEF]=\frac 49 [ABC]. Using this information, we get

[ACFE] = \frac{5}{9}[ABC].

Then, since [ADE] = \frac{1}{9}[ABC], it follows that the [CDEF] = \frac 49 [ABC]. Thus, the answer would be \boxed{\textbf{(A) } \frac{4}{9}}.

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