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Difference between revisions of "2001 AIME II Problems/Problem 7"

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== Problem ==
 
== Problem ==
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Let <math>\triangle{PQR}</math> be a right triangle with <math>PQ = 90</math>, <math>PR = 120</math>, and <math>QR = 150</math>. Let <math>C_{1}</math> be the inscribed circle. Construct <math>\overline{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> is perpendicular to <math>\overline{PR}</math> and tangent to <math>C_{1}</math>. Construct <math>\overline{UV}</math> with <math>U</math> on <math>\overline{PQ}</math> and <math>V</math> on <math>\overline{QR}</math> such that <math>\overline{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <math>\triangle{RST}</math> and <math>C_{3}</math> the inscribed circle of <math>\triangle{QUV}</math>. The distance between the centers of <math>C_{2}</math> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>?
  
 
== Solution ==
 
== Solution ==
 
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== See also ==
 
== See also ==
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* [[2001 AIME II Problems/Problem 6 | Previous problem]]
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{{AIME box|year=2001|n=II|num-b=6|num-a=8}}
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* [[2001 AIME II Problems/Problem 8 | Next problem]]
 
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* [[2001 AIME II Problems]]
 

Revision as of 00:42, 20 November 2007

Problem

Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt {10n}$. What is $n$?

Solution

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See also

2001 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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