Problem #2348

2348.

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]

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

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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