Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎Solution)
m (→‎Solution)
Line 19: Line 19:
 
==Solution==
 
==Solution==
  
−
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>, we can get that the area of region <math>CDEF</math> is <math>\frac{4}{9}=\boxed{\textbf{(B) }\frac{4}{9}}</math> of <math>\triangle ABC</math>.
+
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>, we can get that the area of region <math>CDEF</math> is <math>\frac{4}{9}=\boxed{\textbf{(A) }\frac{4}{9}}</math> of <math>\triangle ABC</math>.
  
 
==See Also==
 
==See Also==

Revision as of 14:44, 22 December 2018

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

The $\triangle ADE$'s area is $\frac{1}{9}$ of $\triangle ABC$ (by similar triangles). We can also get that $\triangle BEF$ is $\frac{4}{9}$ of $\triangle ABC$ (also by similar triangles). Using this information, we can get that the area of region $ACFE$ is $\frac{5}{9}$ of $\triangle ABC$ (because the area of $\triangle BEF$ is $\frac{4}{9}$ of the area of $\triangle ABC$). Then, because $\triangle ADE$ is $\frac{1}{9}$ of $\triangle ABC$, we can get that the area of region $CDEF$ is $\frac{4}{9}=\boxed{\textbf{(A) }\frac{4}{9}}$ of $\triangle ABC$.

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_8_Problems/Problem_20&oldid=99669"