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Difference between revisions of "2011 AIME I Problems/Problem 8"
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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 can be written in the form , where and are relatively prime positive integers and is a positive integer that is not divisible by the square of any prime. Find .
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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>.

Revision as of 21:31, 24 March 2011

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 level 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 $k+m+n$.

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