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Difference between revisions of "2018 AMC 8 Problems/Problem 20"

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==Solution==
 
==Solution==
  
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The <math>\triangle ADE</math>'s area is <math>\frac{1}{9}</math> of <math>\triangle ABC</math> (by similar triangles). We can also get that <math>\triangle BEF</math> is <math>\frac{4}{9}</math> of <math>\triangle ABC</math> (also by similar triangles). Using this information, we can get that the area of region <math>ACFE</math> is <math>\frac{5}{9}</math> of <math>\triangle ABC</math> (because the area of <math>\triangle BEF</math> is <math>\frac{4}{9}</math> of the area of <math>\triangle ABC</math>). Then, because <math>\triangle ADE</math> is <math>\frac{1}{9}</math> of <math>\triangle ABC</math>, it follows that the area of region <math>CDEF</math> is <math>\frac{4}{9}</math><math>\triangle ABC</math>. Thus, the answer would be <math>\boxed {A}.</math>
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By similar triangles, we have <math>[ADE] = \frac{1}{9}[ABC]</math>. Similarly, we see that <math>[BEF] = \frac{4}{9}[ABC].</math> Using this information, we get <cmath>[ACFE] = \frac{5}{9}[ABC].</cmath> Then, since <math>[ADE] = \frac{1}{9}[ABC]</math>, it follows that the <math>[CDEF] = \frac{4}{9}[ABC]</math>. Thus, the answer would be <math>\boxed {A}.</math>
  
 
==See Also==
 
==See Also==

Revision as of 00:29, 3 January 2019

Problem 20

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 $\triangle ABC?$

[asy] size(7cm); pair A,B,C,DD,EE,FF; A = (0,0); B = (3,0); C = (0.5,2.5); EE = (1,0); DD = intersectionpoint(A--C,EE--EE+(C-B)); FF = intersectionpoint(B--C,EE--EE+(C-A)); draw(A--B--C--A--DD--EE--FF,black+1bp); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",DD,W); label("$E$",EE,S); label("$F$",FF,NE); label("$1$",(A+EE)/2,S); label("$2$",(EE+B)/2,S); [/asy]

$\textbf{(A) } \frac{4}{9} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{5}{9} \qquad \textbf{(D) } \frac{3}{5} \qquad \textbf{(E) } \frac{2}{3}$

Solution

By similar triangles, we have $[ADE] = \frac{1}{9}[ABC]$. Similarly, we see that $[BEF] = \frac{4}{9}[ABC].$ Using this information, we get \[[ACFE] = \frac{5}{9}[ABC].\] Then, since $[ADE] = \frac{1}{9}[ABC]$, it follows that the $[CDEF] = \frac{4}{9}[ABC]$. Thus, the answer would be $\boxed {A}.$

See Also

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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