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Difference between revisions of "2008 AIME I Problems/Problem 2"

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Square <math>AIME</math> has sides of length <math>10</math> units.  Isosceles triangle <math>GEM</math> has base <math>EM</math>, and the area common to triangle <math>GEM</math> and square <math>AIME</math> is <math>80</math> square units.  Find the length of the altitude to <math>EM</math> in <math>\triangle GEM</math>.
 
Square <math>AIME</math> has sides of length <math>10</math> units.  Isosceles triangle <math>GEM</math> has base <math>EM</math>, and the area common to triangle <math>GEM</math> and square <math>AIME</math> is <math>80</math> square units.  Find the length of the altitude to <math>EM</math> in <math>\triangle GEM</math>.
  
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== Solution ==
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== Solution 1==
 +
 
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Note that if the altitude of the triangle is at most <math>10</math>, then the maximum area of the intersection of the triangle and the square is <math>5\cdot10=50</math>.
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This implies that vertex G must be located outside of square <math>AIME</math>.
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<center><asy>
 
<center><asy>
 
pair E=(0,0), M=(10,0), I=(10,10), A=(0,10);
 
pair E=(0,0), M=(10,0), I=(10,10), A=(0,10);
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label("\(10\)",(M+E)/2,S);
 
label("\(10\)",(M+E)/2,S);
 
</asy></center>
 
</asy></center>
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Let <math>GE</math> meet <math>AI</math> at <math>X</math> and let <math>GM</math> meet <math>AI</math> at <math>Y</math>. Clearly, <math>XY=6</math> since the area of [[trapezoid]] <math>XYME</math> is <math>80</math>. Also, <math>\triangle GXY \sim \triangle GEM</math>. Let the height of <math>GXY</math> be <math>h</math>.
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Let <math>GE</math> meet <math>AI</math> at <math>X</math> and let <math>GM</math> meet <math>AI</math> at <math>Y</math>. Clearly, <math>XY=6</math> since the area of [[trapezoid]] <math>XYME</math> is <math>80</math>. Also, <math>\triangle GXY \sim \triangle GEM</math>.
  
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By the similarity, <math>\dfrac{h}{6} = \dfrac{h + 10}{10}</math>, we get <math>h = 15</math>. Thus, the height of <math>GEM</math> is <math>h + 10 = \boxed{025}</math>.
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Let the height of <math>GXY</math> be <math>h</math>. By the similarity, <math>\dfrac{h}{6} = \dfrac{h + 10}{10}</math>, we get <math>h = 15</math>. Thus, the height of <math>GEM</math> is <math>h + 10 = \boxed{025}</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}
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this could have been a #1-#5 on the amc 10 lol

Latest revision as of 18:37, 29 December 2020

Problem

Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\triangle GEM$.

Solution 1

Note that if the altitude of the triangle is at most $10$, then the maximum area of the intersection of the triangle and the square is $5\cdot10=50$. This implies that vertex G must be located outside of square $AIME$.

[asy] pair E=(0,0), M=(10,0), I=(10,10), A=(0,10); draw(A--I--M--E--cycle); pair G=(5,25); draw(G--E--M--cycle); label("\(G\)",G,N); label("\(A\)",A,NW); label("\(I\)",I,NE); label("\(M\)",M,NE); label("\(E\)",E,NW); label("\(10\)",(M+E)/2,S); [/asy]

Let $GE$ meet $AI$ at $X$ and let $GM$ meet $AI$ at $Y$. Clearly, $XY=6$ since the area of trapezoid $XYME$ is $80$. Also, $\triangle GXY \sim \triangle GEM$.

Let the height of $GXY$ be $h$. By the similarity, $\dfrac{h}{6} = \dfrac{h + 10}{10}$, we get $h = 15$. Thus, the height of $GEM$ is $h + 10 = \boxed{025}$.

See also

2008 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

this could have been a #1-#5 on the amc 10 lol

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