Difference between revisions of "2019 AIME II Problems/Problem 7"

Revision as of 18:58, 4 November 2022

Problem

Triangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\ell_A,\ell_B$ , and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$ , and $\overline{AB}$ , respectively, such that the intersections of $\ell_A,\ell_B$ , and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_A,\ell_B$ , and $\ell_C$ .

Diagram

$[asy] /* Made by MRENTHUSIASM */ size(350); pair A, B, C, D, e, F, G, H, I, X, Y, Z; B = origin; C = (220,0); A = intersectionpoints(Circle(B,120),Circle(C,180))[0]; D = B+1/4*(C-B); e = C+1/8*(B-C); F = C+1/8*(A-C); G = A+1/4*(C-A); H = A+1/4*(B-A); I = B+1/4*(A-B); X = extension(I,D,F,e); Y = extension(e,F,H,G); Z = extension(G,H,D,I); draw(A--B--C--cycle); draw(X+9/8*(Y-X)--Y+9/8*(X-Y),dashed); draw(Z+9/8*(Y-Z)--Y+9/8*(Z-Y),dashed); draw(X+9/8*(Z-X)--Z+9/8*(X-Z),dashed); draw(H--G^^e--F^^I--D,red); dot("$B$",B,1.5SW,linewidth(4)); dot("$C$",C,1.5SE,linewidth(4)); dot("$A$",A,1.5N,linewidth(4)); dot("$D$",D,1.5SW,linewidth(4)); dot("$E$",e,1.5SE,linewidth(4)); dot("$F$",F,1.5E,linewidth(4)); dot("$G$",G,1.5NE,linewidth(4)); dot("$H$",H,1.5NW,linewidth(4)); dot("$I$",I,1.5W,linewidth(4)); dot("$X$",X,1.5S,linewidth(4)); dot("$Y$",Y,1.5NW,linewidth(4)); dot("$Z$",Z,1.5NE,linewidth(4)); label("$55$",midpoint(H--G),S,red); label("$15$",midpoint(e--F),dir(160),red); label("$45$",midpoint(I--D),dir(55),red); label("$\ell_A$",Z+9/8*(Y-Z),1.5*dir(B--C)); label("$\ell_B$",X+9/8*(Z-X),1.5*dir(C--A)); label("$\ell_C$",Y+9/8*(X-Y),1.5*dir(A--B)); [/asy]$ ~MRENTHUSIASM (diagram edited by integralarefun)

Solution 1

Let the points of intersection of $\ell_A, \ell_B,\ell_C$ with $\triangle ABC$ divide the sides into consecutive segments $BD,DE,EC,CF,FG,GA,AH,HI,IB$ . Furthermore, let the desired triangle be $\triangle XYZ$ , with $X$ closest to side $BC$ , $Y$ closest to side $AC$ , and $Z$ closest to side $AB$ . Hence, the desired perimeter is $XE+EF+FY+YG+GH+HZ+ZI+ID+DX=(DX+XE)+(FY+YG)+(HZ+ZI)+115$ since $HG=55$ , $EF=15$ , and $ID=45$ .

Note that $\triangle AHG\sim \triangle BID\sim \triangle EFC\sim \triangle ABC$ , so using similar triangle ratios, we find that $BI=HA=30$ , $BD=HG=55$ , $FC=\frac{45}{2}$ , and $EC=\frac{55}{2}$ .

We also notice that $\triangle EFC\sim \triangle YFG\sim \triangle EXD$ and $\triangle BID\sim \triangle HIZ$ . Using similar triangles, we get that $FY+YG=\frac{GF}{FC}\cdot \left(EF+EC\right)=\frac{225}{45}\cdot \left(15+\frac{55}{2}\right)=\frac{425}{2}$ $DX+XE=\frac{DE}{EC}\cdot \left(EF+FC\right)=\frac{275}{55}\cdot \left(15+\frac{45}{2}\right)=\frac{375}{2}$ $HZ+ZI=\frac{IH}{BI}\cdot \left(ID+BD\right)=2\cdot \left(45+55\right)=200$ Hence, the desired perimeter is $200+\frac{425+375}{2}+115=600+115=\boxed{715}$ -ktong

Solution 2

Let the diagram be set up like that in Solution 1.

By similar triangles we have $\frac{AH}{AB}=\frac{GH}{BC}\Longrightarrow AH=30$ $\frac{IB}{AB}=\frac{DI}{AC}\Longrightarrow IB=30$ Thus $HI=AB-AH-IB=60$

Since $\bigtriangleup IHZ\sim\bigtriangleup ABC$ and $\frac{HI}{AB}=\frac{1}{2}$ , the altitude of $\bigtriangleup IHZ$ from $Z$ is half the altitude of $\bigtriangleup ABC$ from $C$ , say $\frac{h}{2}$ . Also since $\frac{EF}{AB}=\frac{1}{8}$ , the distance from $\ell_C$ to $AB$ is $\frac{7}{8}h$ . Therefore the altitude of $\bigtriangleup XYZ$ from $Z$ is $\frac{1}{2}h+\frac{7}{8}h=\frac{11}{8}h$ .

By triangle scaling, the perimeter of $\bigtriangleup XYZ$ is $\frac{11}{8}$ of that of $\bigtriangleup ABC$ , or $\frac{11}{8}(220+180+120)=\boxed{715}$

~ Nafer

Solution 3

Notation shown on diagram. By similar triangles we have $k_1 = \frac{EF}{BC} = \frac{AE}{AB} = \frac {AF}{AC} = \frac {1}{4},$ $k_2 = \frac{F''E''}{AC} = \frac {BF''}{AB} = \frac{1}{4},$ $k_3 = \frac{E'F'}{AB} = \frac{E'C }{AC} = \frac{1}{8}.$ So, $\frac{ZE}{BC} = \frac{F''E}{AB} = \frac{AB - AE - BF''}{AB} = 1 - k_1 - k_2,$ $\frac{FY}{BC} = \frac{FE'}{AC} = \frac{AC - AF - CE'}{AC} = 1 - k_1 - k_3.$ $k = \frac{ZY}{BC} = \frac{ZE + EF + FY}{BC} = (1 - k_1 - k_2) + k_1 + (1 - k_1 - k_3)$ $k = 2 - k_1 - k_2 - k_3 = 2 - \frac{1}{4} - \frac{1}{4} - \frac{1}{8} = \frac{11}{8}.$ $\frac{ZY+YX +XZ}{BC +AB + AC} = k \implies ZY + YX + XZ =\frac{11}{8} (220 + 120 + 180) = \boxed {715}.$ vladimir.shelomovskii@gmail.com, vvsss

2019 AIME II (Problems • Answer Key • Resources)
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