{"status": "success", "data": {"description_md": "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 $PQ$?<br><center><img class=\"problem-image\" alt='[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]' class=\"latexcenter\" height=\"442\" src=\"https://latex.artofproblemsolving.com/c/7/7/c77d6149eac683a6a20552ee73fb3c8fe5638b07.png\" width=\"385\"/></center>\n\n$\\textbf{(A)}\\ 1 \\qquad<br>\\textbf{(B)}\\ \\frac{5}{8}\\sqrt{3} \\qquad<br>\\textbf{(C)}\\ \\frac{4}{5}\\sqrt{2} \\qquad<br>\\textbf{(D)}\\ \\frac{8}{15}\\sqrt{5} \\qquad<br>\\textbf{(E)}\\ \\frac{6}{5}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>In  <span class=\"katex--inline\">\\triangle ABC</span>  shown in the figure,  <span class=\"katex--inline\">AB=7</span> ,  <span class=\"katex--inline\">BC=8</span> ,  <span class=\"katex--inline\">CA=9</span> , and  <span class=\"katex--inline\">\\overline{AH}</span>  is an altitude. Points  <span class=\"katex--inline\">D</span>  and  <span class=\"katex--inline\">E</span>  lie on sides  <span class=\"katex--inline\">\\overline{AC}</span>  and  <span class=\"katex--inline\">\\overline{AB}</span> , respectively, so that  <span class=\"katex--inline\">\\overline{BD}</span>  and  <span class=\"katex--inline\">\\overline{CE}</span>  are angle bisectors, intersecting  <span class=\"katex--inline\">\\overline{AH}</span>  at  <span class=\"katex--inline\">Q</span>  and  <span class=\"katex--inline\">P</span> , respectively. What is  <span class=\"katex--inline\">PQ</span> ?<br/><center><img class=\"latexcenter\" alt=\"[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(&#34;7&#34;,(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor);  label(&#34;9&#34;,(2.0037562070503783,4.196493361737088),SE*labelscalefactor);  label(&#34;8&#34;,(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(&#34;$A$&#34;, (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor);  dot((7.,0.),dotstyle);  label(&#34;$B$&#34;, (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor);  dot((3.7058823529411766,0.),dotstyle);  label(&#34;$E$&#34;, (3.8123296394941084,0.16830708038513573), NE * labelscalefactor);  dot((4.714285714285714,7.666518779999279),dotstyle);  label(&#34;$C$&#34;, (4.579603216894479,7.895848109917452), NE * labelscalefactor);  dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle);  label(&#34;$D$&#34;, (2.1407693458718726,3.127790878929427), NE * labelscalefactor);  dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle);  label(&#34;$H$&#34;, (6.004539860638023,1.9494778850645704), NE * labelscalefactor);  dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle);  label(&#34;$Q$&#34;, (4.935837377830365,1.7302568629501784), NE * labelscalefactor);  dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle);  label(&#34;$P$&#34;, (3.538303361851119,1.2370095631927964), NE * labelscalefactor);  clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);   /* end of picture */ [/asy]\" height=\"442\" src=\"https://latex.artofproblemsolving.com/c/7/7/c77d6149eac683a6a20552ee73fb3c8fe5638b07.png\" width=\"385\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\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}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2016 AMC 12B Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/16_amc12B_p18", "prev": "/problem/16_amc12B_p16"}}