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Difference between revisions of "2011 AIME I Problems/Problem 8"

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== Problem ==
 
== Problem ==
−
In triangle <math>ABC</math>, <math>BC = 23</math>, <math>CA = 27</math>, and <math>AB = 30</math>. Points <math>V</math> and <math>W</math> are on <math>\overline{AC}</math> with <math>V</math> on <math> \overline{AW} </math>, points <math>X</math> and <math>Y</math> are on <math> \overline{BC} </math> with <math>X</math> on <math> \overline{CY} </math>, and points <math>Z</math> and <math>U</math> are on <math> \overline{AB} </math> with <math>Z</math> on <math> \overline{BU} </math>. In addition, the points are positioned so that <math> \overline{UV}\parallel\overline{BC} </math>, <math> \overline{WX}\parallel\overline{AB} </math>, and <math> \overline{YZ}\parallel\overline{CA} </math>. Right angle folds are then made along <math> \overline{UV} </math>, <math> \overline{WX} </math>, and <math> \overline{YZ} </math>. The resulting figure is placed on a level floor to make a table with triangular legs. Let <math>h</math> be the maximum possible height of a table constructed from triangle <math>ABC</math> whose top is parallel to the floor. Then <math>h</math> can be written in the form <math> \frac{k\sqrt{m}}{n} </math>, where <math>k</math> and <math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <math>k+m+n</math>.
+
In triangle <math>ABC</math>, <math>BC = 23</math>, <math>CA = 27</math>, and <math>AB = 30</math>. Points <math>V</math> and <math>W</math> are on <math>\overline{AC}</math> with <math>V</math> on <math> \overline{AW} </math>, points <math>X</math> and <math>Y</math> are on <math> \overline{BC} </math> with <math>X</math> on <math> \overline{CY} </math>, and points <math>Z</math> and <math>U</math> are on <math> \overline{AB} </math> with <math>Z</math> on <math> \overline{BU} </math>. In addition, the points are positioned so that <math> \overline{UV}\parallel\overline{BC} </math>, <math> \overline{WX}\parallel\overline{AB} </math>, and <math> \overline{YZ}\parallel\overline{CA} </math>. Right angle folds are then made along <math> \overline{UV} </math>, <math> \overline{WX} </math>, and <math> \overline{YZ} </math>. The resulting figure is placed on a leveled floor to make a table with triangular legs. Let <math>h</math> be the maximum possible height of a table constructed from triangle <math>ABC</math> whose top is parallel to the floor. Then <math>h</math> can be written in the form <math> \frac{k\sqrt{m}}{n} </math>, where <math>k</math> and <math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <math>k+m+n</math>.
  
  
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dot("$Y$",Y[1],dir(250));
 
dot("$Y$",Y[1],dir(250));
 
dot("$Z$",Z[1],NE);</asy></center>
 
dot("$Z$",Z[1],NE);</asy></center>
 +
 +
==Solution 1==
 +
Note that triangles <math>\triangle AUV, \triangle BYZ</math> and <math>\triangle CWX</math> all have the same height because when they are folded up to create the legs of the table, the top needs to be parallel to the floor. We want to find the maximum possible value of this height, given that no two of <math>\overline{UV}, \overline{WX}</math> and <math>\overline{YZ}</math> intersect inside <math>\triangle ABC</math>. Let <math>h_{A}</math> denote the length of the altitude dropped from vertex <math>A,</math> and define <math>h_{B}</math> and <math>h_{C}</math> similarly. Also let <math>\{u, v, w, x, y, z\} = \{AU, AV, CW, CX, BY, BZ\}</math>. Then by similar triangles
 +
<cmath>\begin{align}
 +
\frac{u}{AB}=\frac{v}{AC}=\frac{h}{h_{A}}, \\
 +
\frac{w}{CA}=\frac{x}{CB}=\frac{h}{h_{C}}, \\
 +
\frac{y}{BC}=\frac{z}{BA}=\frac{h}{h_{B}}.
 +
\end{align}</cmath>
 +
Since <math>h_{A}=\frac{2K}{23}</math> and similarly for <math>27</math> and <math>30,</math> where <math>K</math> is the area of <math>\triangle ABC,</math> we can write
 +
<cmath>\begin{align}
 +
\frac{u}{30}=\frac{v}{27}=\frac{h}{\tfrac{2K}{23}}, \\
 +
\frac{w}{27}=\frac{x}{23}=\frac{h}{\tfrac{2K}{30}}, \\
 +
\frac{y}{23}=\frac{z}{30}=\frac{h}{\tfrac{2K}{27}}.
 +
\end{align}</cmath>
 +
and simplifying gives <math>u=x=\frac{690h}{2K}, v=y=\frac{621h}{2K}, w=z=\frac{810h}{2K}</math>. Because no two segments can intersect inside the triangle, we can form the inequalities <math>v+w\leq 27, x+y\leq 23,</math> and <math>z+u\leq 30</math>. That is, all three of the inequalities
 +
<cmath>\begin{align}
 +
\frac{621h+810h}{2K}\leq 27, \\
 +
\frac{690h+621h}{2K}\leq 23, \\
 +
\frac{810h+690h}{2K}\leq 30.
 +
\end{align}</cmath>
 +
must hold. Dividing both sides of each equation by the RHS, we have
 +
<cmath>\begin{align}
 +
\frac{53h}{2K}\leq 1\, \text{since}\, \frac{1431}{27}=53, \\
 +
\frac{57h}{2K}\leq 1\, \text{since}\, \frac{1311}{23}=57, \\
 +
\frac{50h}{2K}\leq 1\, \text{since}\, \frac{1500}{30}=50.
 +
\end{align}</cmath>
 +
It is relatively easy to see that <math>\frac{57h}{2K}\leq 1</math> restricts us the most since it cannot hold if the other two do not hold. The largest possible value of <math>h</math> is thus <math>\frac{2K}{57},</math> and note that by Heron's formula the area of <math>\triangle ABC</math> is <math>20\sqrt{221}</math>. Then <math>\frac{2K}{57}=\frac{40\sqrt{221}}{57},</math> and the answer is <math>40+221+57=261+57=\boxed{318}</math>
 +
 +
~sugar_rush
 +
 +
==Solution 2==
 +
Note that the area is given by Heron's formula and it is <math>20\sqrt{221}</math>. Let <math>h_i</math> denote the length of the altitude dropped from vertex i. It follows that <math>h_b = \frac{40\sqrt{221}}{27}, h_c  = \frac{40\sqrt{221}}{30}, h_a = \frac{40\sqrt{221}}{23}</math>. From similar triangles we can see that <math>\frac{27h}{h_a}+\frac{27h}{h_c} \le 27 \rightarrow h \le \frac{h_ah_c}{h_a+h_c}</math>. We can see this is true for any combination of a,b,c and thus the minimum of the upper bounds for h yields <math>h = \frac{40\sqrt{221}}{57} \rightarrow \boxed{318}</math>.
 +
 +
==Solution 3==
 +
 +
As from above, we can see that the length of the altitude from A is the longest. Thus the highest table is formed when X and Y meet up. Let the distance of this point from B be <math>x</math>, making the distance from C <math>23 - x</math>. Let <math>h</math> be the height of the table. From similar triangles, we have <math>\frac{x}{23} = \frac{h}{h_b} = \frac{27h}{2A}</math> where A is the area of ABC. Similarly, <math>\frac{23-x}{23}=\frac{h}{h_c}=\frac{30h}{2A}</math>. Therefore, <math>1-\frac{x}{23}=\frac{30h}{2A} \rightarrow1-\frac{27h}{2A}=\frac{30h}{2A}</math> and hence <math> h = \frac{2A}{57} = \frac{40\sqrt{221}}{57}\rightarrow \boxed{318}</math>.
 +
 +
==Video Solution by Punxsutawney Phil==
 +
https://youtube.com/watch?v=Asvy1s-6Rx0
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2011|n=I|num-b=7|num-a=9}}
 
{{AIME box|year=2011|n=I|num-b=7|num-a=9}}
 +
 +
[[Category:Intermediate Geometry Problems]]
 +
[[Category:3D Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 20:47, 12 April 2022

Problem

In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that $\overline{UV}\parallel\overline{BC}$, $\overline{WX}\parallel\overline{AB}$, and $\overline{YZ}\parallel\overline{CA}$. Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$. The resulting figure is placed on a leveled floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\frac{k\sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k+m+n$.


[asy] unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE);[/asy]

Solution 1

Note that triangles $\triangle AUV, \triangle BYZ$ and $\triangle CWX$ all have the same height because when they are folded up to create the legs of the table, the top needs to be parallel to the floor. We want to find the maximum possible value of this height, given that no two of $\overline{UV}, \overline{WX}$ and $\overline{YZ}$ intersect inside $\triangle ABC$. Let $h_{A}$ denote the length of the altitude dropped from vertex $A,$ and define $h_{B}$ and $h_{C}$ similarly. Also let $\{u, v, w, x, y, z\} = \{AU, AV, CW, CX, BY, BZ\}$. Then by similar triangles \begin{align} \frac{u}{AB}=\frac{v}{AC}=\frac{h}{h_{A}}, \\ \frac{w}{CA}=\frac{x}{CB}=\frac{h}{h_{C}}, \\ \frac{y}{BC}=\frac{z}{BA}=\frac{h}{h_{B}}. \end{align} Since $h_{A}=\frac{2K}{23}$ and similarly for $27$ and $30,$ where $K$ is the area of $\triangle ABC,$ we can write \begin{align} \frac{u}{30}=\frac{v}{27}=\frac{h}{\tfrac{2K}{23}}, \\ \frac{w}{27}=\frac{x}{23}=\frac{h}{\tfrac{2K}{30}}, \\ \frac{y}{23}=\frac{z}{30}=\frac{h}{\tfrac{2K}{27}}. \end{align} and simplifying gives $u=x=\frac{690h}{2K}, v=y=\frac{621h}{2K}, w=z=\frac{810h}{2K}$. Because no two segments can intersect inside the triangle, we can form the inequalities $v+w\leq 27, x+y\leq 23,$ and $z+u\leq 30$. That is, all three of the inequalities \begin{align} \frac{621h+810h}{2K}\leq 27, \\ \frac{690h+621h}{2K}\leq 23, \\ \frac{810h+690h}{2K}\leq 30. \end{align} must hold. Dividing both sides of each equation by the RHS, we have \begin{align} \frac{53h}{2K}\leq 1\, \text{since}\, \frac{1431}{27}=53, \\ \frac{57h}{2K}\leq 1\, \text{since}\, \frac{1311}{23}=57, \\ \frac{50h}{2K}\leq 1\, \text{since}\, \frac{1500}{30}=50. \end{align} It is relatively easy to see that $\frac{57h}{2K}\leq 1$ restricts us the most since it cannot hold if the other two do not hold. The largest possible value of $h$ is thus $\frac{2K}{57},$ and note that by Heron's formula the area of $\triangle ABC$ is $20\sqrt{221}$. Then $\frac{2K}{57}=\frac{40\sqrt{221}}{57},$ and the answer is $40+221+57=261+57=\boxed{318}$

~sugar_rush

Solution 2

Note that the area is given by Heron's formula and it is $20\sqrt{221}$. Let $h_i$ denote the length of the altitude dropped from vertex i. It follows that $h_b = \frac{40\sqrt{221}}{27}, h_c = \frac{40\sqrt{221}}{30}, h_a = \frac{40\sqrt{221}}{23}$. From similar triangles we can see that $\frac{27h}{h_a}+\frac{27h}{h_c} \le 27 \rightarrow h \le \frac{h_ah_c}{h_a+h_c}$. We can see this is true for any combination of a,b,c and thus the minimum of the upper bounds for h yields $h = \frac{40\sqrt{221}}{57} \rightarrow \boxed{318}$.

Solution 3

As from above, we can see that the length of the altitude from A is the longest. Thus the highest table is formed when X and Y meet up. Let the distance of this point from B be $x$, making the distance from C $23 - x$. Let $h$ be the height of the table. From similar triangles, we have $\frac{x}{23} = \frac{h}{h_b} = \frac{27h}{2A}$ where A is the area of ABC. Similarly, $\frac{23-x}{23}=\frac{h}{h_c}=\frac{30h}{2A}$. Therefore, $1-\frac{x}{23}=\frac{30h}{2A} \rightarrow1-\frac{27h}{2A}=\frac{30h}{2A}$ and hence $h = \frac{2A}{57} = \frac{40\sqrt{221}}{57}\rightarrow \boxed{318}$.

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=Asvy1s-6Rx0

See also

2011 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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