MathDB
New Coord Bash Problem

Source: 2016 AMC 10B #19

February 21, 2016
AMCAMC 10AMC 10 BAMC 12geometry2016 AMC 10B

Problem Statement

Rectangle ABCDABCD has AB=5AB=5 and BC=4BC=4. Point EE lies on AB\overline{AB} so that EB=1EB=1, point GG lies on BC\overline{BC} so that CG=1CG=1. and point FF lies on CD\overline{CD} so that DF=2DF=2. Segments AG\overline{AG} and AC\overline{AC} intersect EF\overline{EF} at QQ and PP, respectively. What is the value of PQEF\frac{PQ}{EF}?
[asy] pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label("QQ",(3.07692307692,2.15384615384),N); label("PP",(20/7,12/7),W); label("AA",(0,4), NW); label("BB",(5,4), NE); label("CC",(5,0),SE); label("DD",(0,0),SW); label("FF",(2,0),S); label("GG",(5,1),E); label("EE",(4,4),N);
dot(A1); dot(A2); dot(B1); dot(B2); dot(C1); dot(C2); dot((0,0)); dot((5,4));[/asy]
<spanclass=latexbold>(A)</span> 1316<spanclass=latexbold>(B)</span> 213<spanclass=latexbold>(C)</span> 982<spanclass=latexbold>(D)</span> 1091<spanclass=latexbold>(E)</span> 19<span class='latex-bold'>(A)</span>~\frac{\sqrt{13}}{16} \qquad <span class='latex-bold'>(B)</span>~\frac{\sqrt{2}}{13} \qquad <span class='latex-bold'>(C)</span>~\frac{9}{82} \qquad <span class='latex-bold'>(D)</span>~\frac{10}{91}\qquad <span class='latex-bold'>(E)</span>~\frac19