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Difference between revisions of "2016 AMC 10B Problems/Problem 19"

(Created page with "==Problem== Rectangle <math>ABCD</math> has <math>AB=5</math> and <math>BC=4</math>. Point <math>E</math> lies on line <math>AB</math> so that <math>EB=1</math>, point <math>...")
 
(→‎Problem)
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==Problem==
 
==Problem==
  
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Rectangle <math>ABCD</math> has <math>AB=5</math> and <math>BC=4</math>. Point <math>E</math> lies on line <math>AB</math> so that <math>EB=1</math>, point <math>G</math> lies on line <math>BC</math> so that <math>CG=1</math>, and point <math>F</math> lies on line <math>CD</math> so that <math>DF=2</math>. Segments <math>AG</math> and <math>AC</math> intersect <math>EF</math> at <math>Q</math> and <math>P</math>, respectively. What is the value of <math>\frac{PQ}{EF}?</math>
+
Rectangle <math>ABCD</math> has <math>AB=5</math> and <math>BC=4</math>. Point <math>E</math> lies on <math>\overline{AB}</math> so that <math>EB=1</math>, point <math>G</math> lies on <math>\overline{BC}</math> so that <math>CG=1</math>. and point <math>F</math> lies on <math>\overline{CD}</math> so that <math>DF=2</math>. Segments <math>\overline{AG}</math> and <math>\overline{AC}</math> intersect <math>\overline{EF}</math> at <math>Q</math> and <math>P</math>, respectively. What is the value of <math>\frac{PQ}{EF}</math>?
  
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<cmath>(A) \frac{\sqrt{3}}{16}, (B) \frac{\sqrt{2}}{13}, (C), \frac{9}{82}, (D) \frac{10}{91}, (E) \frac{1}{9}</cmath>
+
 
 +
[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("<math>Q</math>",(3.07692307692,2.15384615384),N);
 +
label("<math>P</math>",(20/7,12/7),W);
 +
label("<math>A</math>",(0,4), NW);
 +
label("<math>B</math>",(5,4), NE);
 +
label("<math>C</math>",(5,0),SE);
 +
label("<math>D</math>",(0,0),SW);
 +
label("<math>F</math>",(2,0),S); label("<math>G</math>",(5,1),E);
 +
label("<math>E</math>",(4,4),N);[/asy]
 +
 
 +
<math>\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad
 +
\textbf{(B)}~\frac{\sqrt{2}}{13} \qquad
 +
\textbf{(C)}~\frac{9}{82} \qquad
 +
\textbf{(D)}~\frac{10}{91}\qquad
 +
\textbf{(E)}~\frac19</math>

Revision as of 09:31, 21 February 2016

Problem

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?


[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("$Q$",(3.07692307692,2.15384615384),N); label("$P$",(20/7,12/7),W); label("$A$",(0,4), NW); label("$B$",(5,4), NE); label("$C$",(5,0),SE); label("$D$",(0,0),SW); label("$F$",(2,0),S); label("$G$",(5,1),E); label("$E$",(4,4),N);[/asy]

$\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad \textbf{(B)}~\frac{\sqrt{2}}{13} \qquad \textbf{(C)}~\frac{9}{82} \qquad \textbf{(D)}~\frac{10}{91}\qquad \textbf{(E)}~\frac19$

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