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

Latest revision as of 19:12, 23 October 2023

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 1

Get the area of the triangle by Heron's Formula: $\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{(12)(3)(4)(5)} = 12\sqrt{5}$ Use the area to find the height $AH$ with known base $BC$ : $Area = 12\sqrt{5} = \frac{1}{2}bh = \frac{1}{2}(8)(AH)$ $AH = 3\sqrt{5}$ $BH = \sqrt{AB^2 - AH^2} = \sqrt{7^2 - (3\sqrt{5})^2} = 2$ $CH = BC - BH = 8 - 2 = 6$ Apply the Angle Bisector Theorem on $\triangle ACH$ and $\triangle ABH$ , we get $AP:PH = 9:6$ and $AQ:QH = 7:2$ , respectively. To find $AP$ , $PH$ , $AQ$ , and $QH$ , apply variables, such that $AP:PH = 9:6$ is $\frac{3\sqrt{5} - x}{x} = \frac{9}{6}$ and $AQ:QH = 7:2$ is $\frac{3\sqrt{5} - y}{y} = \frac{7}{2}$ . Solving them out, you will get $AP = \frac{9\sqrt{5}}{5}$ , $PH = \frac{6\sqrt{5}}{5}$ , $AQ = \frac{7\sqrt{5}}{3}$ , and $QH = \frac{2\sqrt{5}}{3}$ . Then, since $AP + PQ = AQ$ according to the Segment Addition Postulate, and thus manipulating, you get $PQ = AQ - AP = \frac{7\sqrt{5}}{3} - \frac{9\sqrt{5}}{5}$ = $\boxed{\textbf{(D)}\frac{8}{15}\sqrt{5}}$

Solution 2

Let the intersection of $BD$ and $CE$ be the point $I$ . Then let the foot of the altitude from $I$ to $BC$ be $I'$ . Note that $II'$ is an inradius and that $II' \cdot s = [ABC]$ , where $s$ is the semiperimeter of the triangle.

Using Heron's Formula, we see that $II' \cdot 12 = \sqrt{12 \cdot 3 \cdot 4 \cdot 5} = 12\sqrt{5}$ , so $II' = \sqrt{5}$ .

Then since $II'$ and $AH$ are parallel, $\triangle CI'I \sim \triangle CHP$ and $\triangle BHQ \sim \triangle BI'I$ .

Thus, $\frac{II'}{PQ + QH} = \frac{CI'}{CH}$ and $\frac{II'}{QH} = \frac{BI'}{BH}$ , so $PQ = \frac{II' \cdot CH}{CI'} - \frac{II' \cdot BH}{BI'}$ .

By the Dual Principle, $CI' = 5$ and $BI' = 3$ . With the same method as Solution 1, $CH = 6$ and $BH = 2$ . Then $PQ = \frac{8}{15} II' =$ $\boxed{\textbf{(D)}\frac{8}{15}\sqrt{5}}$

Solution 3 (FAST)

$PQ$ lies on altitude $AH$ , which we find to have a length of $3\sqrt{5}$ by Heron's Formula and dividing twice the area by $BC$ . From H we can construct a segment $HX$ with $X$ on $CE$ such that $HX$ is parallel to $EB$ . A similar construction gives $Y$ on $BD$ such that $HY$ is parallel to $DC$ . We can hence generate a system of ratios that will allow us to find $PQ/AH$ . Note that such a system will generate a rational number for the ratio $PQ/AH$ . Thus, we choose the only answer that has a $\sqrt{5}$ term in it, giving us $\boxed{\textbf{(D)}}$ .

Solution 4

Let $h=AH$ and $BH=x$ . Then, $CH=8-x$ . By the Pythagorean Theorem on right triangles $ABH$ and $ACH$ , we have $h^2+x^2=49$ $x^2+(8-x)^2=81.$ Subtracting the prior from the latter yields $-16x+64=32\implies x=2$ . So, $BH=2$ , $CH=6$ , and $AH=3\sqrt{5}$ . Continue with Solution 1.

Video Solution

https://www.youtube.com/watch?v=ccB-z4_OHqw

@@ Line 49: / Line 49: @@
 \textbf{(E)}\ \frac{6}{5}</math>
-==Solution==
+==Solution 1==
+Get the area of the triangle by [https://artofproblemsolving.com/wiki/index.php/Heron%27s_Formula Heron's Formula]:
+<cmath>\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{(12)(3)(4)(5)} = 12\sqrt{5}</cmath>
+Use the area to find the height <math>AH</math> with known base <math>BC</math>:
+<cmath>Area = 12\sqrt{5} = \frac{1}{2}bh = \frac{1}{2}(8)(AH)</cmath>
+<cmath>AH = 3\sqrt{5}</cmath>
+<cmath>BH = \sqrt{AB^2 - AH^2} = \sqrt{7^2 - (3\sqrt{5})^2} = 2</cmath>
+<cmath>CH = BC - BH = 8 - 2 = 6</cmath>
+Apply the [https://artofproblemsolving.com/wiki/index.php/Angle_Bisector_Theorem Angle Bisector Theorem] on <math>	\triangle ACH</math> and <math>\triangle ABH</math>, we get <math>AP:PH = 9:6</math> and <math>AQ:QH = 7:2</math>, respectively.
+To find <math>AP</math>, <math>PH</math>, <math>AQ</math>, and <math>QH</math>, apply variables, such that <math>AP:PH = 9:6</math> is <math>\frac{3\sqrt{5} - x}{x} = \frac{9}{6}</math> and <math>AQ:QH = 7:2</math> is <math>\frac{3\sqrt{5} - y}{y} = \frac{7}{2}</math>. Solving them out, you will get <math>AP = \frac{9\sqrt{5}}{5}</math>, <math>PH = \frac{6\sqrt{5}}{5}</math>, <math>AQ = \frac{7\sqrt{5}}{3}</math>, and <math>QH = \frac{2\sqrt{5}}{3}</math>. Then, since <math>AP + PQ = AQ</math> according to the Segment Addition Postulate, and thus manipulating, you get <math>PQ = AQ - AP = \frac{7\sqrt{5}}{3} - \frac{9\sqrt{5}}{5}</math> = <cmath>\boxed{\textbf{(D)}\frac{8}{15}\sqrt{5}}</cmath>
+==Solution 2==
+Let the intersection of <math>BD</math> and <math>CE</math> be the point <math>I</math>. Then let the foot of the altitude from <math>I</math> to <math>BC</math> be <math>I'</math>. Note that <math>II'</math> is an inradius and that <math>II' \cdot s = [ABC]</math>, where <math>s</math> is the semiperimeter of the triangle.
+Using Heron's Formula, we see that <math>II' \cdot 12 =  \sqrt{12 \cdot 3 \cdot 4 \cdot 5} = 12\sqrt{5}</math>, so <math>II' = \sqrt{5}</math>.
+Then since <math>II'</math> and <math>AH</math> are parallel, <math>\triangle CI'I \sim \triangle CHP</math> and <math>\triangle BHQ \sim \triangle BI'I</math>.
+Thus, <math>\frac{II'}{PQ + QH} = \frac{CI'}{CH}</math> and <math>\frac{II'}{QH} = \frac{BI'}{BH}</math>, so
+<math>PQ = \frac{II' \cdot CH}{CI'} - \frac{II' \cdot BH}{BI'}</math>.
+By the Dual Principle, <math>CI' = 5</math> and <math>BI' = 3</math>. With the same method as Solution 1, <math>CH = 6</math> and <math>BH = 2</math>.
+Then <math>PQ  = \frac{8}{15} II' = </math> <cmath>\boxed{\textbf{(D)}\frac{8}{15}\sqrt{5}}</cmath>
+==Solution 3 (FAST)==
+<math>PQ</math> lies on altitude <math>AH</math>, which we find to have a length of <math>3\sqrt{5}</math> by Heron's Formula and dividing twice the area by <math>BC</math>. From H we can construct a segment <math>HX</math> with <math>X</math> on <math>CE</math> such that <math>HX</math> is parallel to <math>EB</math>. A similar construction gives <math>Y</math> on <math>BD</math> such that <math>HY</math> is parallel to <math>DC</math>. We can hence generate a system of ratios that will allow us to find <math>PQ/AH</math>. Note that such a system will generate a rational number for the ratio <math>PQ/AH</math>. Thus, we choose the only answer that has a <math>\sqrt{5}</math> term in it, giving us <math>\boxed{\textbf{(D)}} </math>.
+==Solution 4 ==
+Let <math>h=AH</math> and <math>BH=x</math>.  Then, <math>CH=8-x</math>.  By the Pythagorean Theorem on right triangles <math>ABH</math> and <math>ACH</math>, we have <cmath>h^2+x^2=49</cmath> <cmath>x^2+(8-x)^2=81.</cmath>  Subtracting the prior from the latter yields <math>-16x+64=32\implies x=2</math>.  So, <math>BH=2</math>, <math>CH=6</math>, and <math>AH=3\sqrt{5}</math>.  Continue with Solution 1.
+==Video Solution==
+https://www.youtube.com/watch?v=ccB-z4_OHqw
 ==See Also==
 {{AMC12 box|year=2016|ab=B|num-b=16|num-a=18}}
 {{MAA Notice}}

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