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Difference between revisions of "2016 AMC 12B Problems/Problem 17"

(Created page with "==Problem== In <math>\triangle ABC</math> shown in the figure, <math>AB=7</math>, <math>BC=8</math>, <math>CA=9</math>, and <math>\overline{AH}</math> is an altitude. Points ...")
 
Line 3: Line 3:
 
In <math>\triangle ABC</math> shown in the figure, <math>AB=7</math>, <math>BC=8</math>, <math>CA=9</math>, and <math>\overline{AH}</math> is an altitude. Points <math>D</math> and <math>E</math> lie on sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively, so that <math>\overline{BD}</math> and <math>\overline{CE}</math> are angle bisectors, intersecting <math>\overline{AH}</math> at <math>Q</math> and <math>P</math>, respectively. What is <math>PQ</math>?
 
In <math>\triangle ABC</math> shown in the figure, <math>AB=7</math>, <math>BC=8</math>, <math>CA=9</math>, and <math>\overline{AH}</math> is an altitude. Points <math>D</math> and <math>E</math> lie on sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively, so that <math>\overline{BD}</math> and <math>\overline{CE}</math> are angle bisectors, intersecting <math>\overline{AH}</math> at <math>Q</math> and <math>P</math>, respectively. What is <math>PQ</math>?
  
−
[asy]
+
<asy>
 
import graph; size(9cm);  
 
import graph; size(9cm);  
 
real labelscalefactor = 0.5; /* changes label-to-point distance */
 
real labelscalefactor = 0.5; /* changes label-to-point distance */
Line 24: Line 24:
 
  /* dots and labels */
 
  /* dots and labels */
 
dot((0.,0.),dotstyle);  
 
dot((0.,0.),dotstyle);  
−
label("<math>A</math>", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor);  
+
label("$A$", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor);  
 
dot((7.,0.),dotstyle);  
 
dot((7.,0.),dotstyle);  
−
label("<math>B</math>", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor);  
+
label("$B$", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor);  
 
dot((3.7058823529411766,0.),dotstyle);  
 
dot((3.7058823529411766,0.),dotstyle);  
−
label("<math>E</math>", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor);  
+
label("$E$", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor);  
 
dot((4.714285714285714,7.666518779999279),dotstyle);  
 
dot((4.714285714285714,7.666518779999279),dotstyle);  
−
label("<math>C</math>", (4.579603216894479,7.895848109917452), NE * labelscalefactor);  
+
label("$C$", (4.579603216894479,7.895848109917452), NE * labelscalefactor);  
 
dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle);  
 
dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle);  
−
label("<math>D</math>", (2.1407693458718726,3.127790878929427), NE * labelscalefactor);  
+
label("$D$", (2.1407693458718726,3.127790878929427), NE * labelscalefactor);  
 
dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle);  
 
dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle);  
−
label("<math>H</math>", (6.004539860638023,1.9494778850645704), NE * labelscalefactor);  
+
label("$H$", (6.004539860638023,1.9494778850645704), NE * labelscalefactor);  
 
dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle);  
 
dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle);  
−
label("<math>Q</math>", (4.935837377830365,1.7302568629501784), NE * labelscalefactor);  
+
label("$Q$", (4.935837377830365,1.7302568629501784), NE * labelscalefactor);  
 
dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle);  
 
dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle);  
−
label("<math>P</math>", (3.538303361851119,1.2370095631927964), NE * labelscalefactor);  
+
label("$P$", (3.538303361851119,1.2370095631927964), NE * labelscalefactor);  
 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
  /* end of picture */
 
  /* end of picture */
−
[/asy]
+
</asy>
  
 
<math>\textbf{(A)}\ 1 \qquad
 
<math>\textbf{(A)}\ 1 \qquad

Revision as of 11:13, 21 February 2016

Problem

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?

[asy] import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.381056062031275, xmax = 15.020004395092375, ymin = -4.051697595316909, ymax = 10.663513514111651; /* image dimensions */ draw((0.,0.)--(4.714285714285714,7.666518779999279)--(7.,0.)--cycle); /* draw figures */ draw((0.,0.)--(4.714285714285714,7.666518779999279)); draw((4.714285714285714,7.666518779999279)--(7.,0.)); draw((7.,0.)--(0.,0.)); label("7",(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor); label("9",(2.0037562070503783,4.196493361737088),SE*labelscalefactor); label("8",(6.114150371695219,3.785453945272603),SE*labelscalefactor); draw((0.,0.)--(6.428571428571427,1.9166296949998194)); draw((7.,0.)--(2.2,3.5777087639996634)); draw((4.714285714285714,7.666518779999279)--(3.7058823529411766,0.)); /* dots and labels */ dot((0.,0.),dotstyle); label("$A$", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor); dot((7.,0.),dotstyle); label("$B$", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor); dot((3.7058823529411766,0.),dotstyle); label("$E$", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor); dot((4.714285714285714,7.666518779999279),dotstyle); label("$C$", (4.579603216894479,7.895848109917452), NE * labelscalefactor); dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle); label("$D$", (2.1407693458718726,3.127790878929427), NE * labelscalefactor); dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle); label("$H$", (6.004539860638023,1.9494778850645704), NE * labelscalefactor); dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle); label("$Q$", (4.935837377830365,1.7302568629501784), NE * labelscalefactor); dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle); label("$P$", (3.538303361851119,1.2370095631927964), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad \textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad \textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad \textbf{(E)}\ \frac{6}{5}$

Solution

See Also

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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 AMC 12 Problems and Solutions

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