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Difference between revisions of "2006 iTest Problems/Problem U7"

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==Problem==
 
==Problem==
  
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Let  <math>T  =  TNFTPP</math>. Triangle  <math>ABC</math> has integer side lengths, including  <math>BC  =  100T  -  4</math>, and a right angle,  <math>\angle ABC</math>. Let  <math>r</math> and  <math>s</math> denote the inradius and semiperimeter of  <math>ABC</math> respectively. Find the perimeter of the triangle ABC which minimizes  <math>\frac{s}{r}</math>.
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Let  <math>T  =  TNFTPP</math>. Triangle  <math>ABC</math> has integer side lengths, including  <math>BC  =  696</math>, and a right angle,  <math>\angle ABC</math>. Let  <math>r</math> and  <math>s</math> denote the inradius and semiperimeter of  <math>ABC</math> respectively. Find the perimeter of the triangle ABC which minimizes  <math>\frac{s}{r}</math>.
  
 
==Solution==
 
==Solution==
 
First, label the other leg <math>x</math> and the hypotenuse <math>y</math>. To minimize
 
First, label the other leg <math>x</math> and the hypotenuse <math>y</math>. To minimize

Revision as of 21:37, 17 November 2019

Problem

Let $T = TNFTPP$. Triangle $ABC$ has integer side lengths, including $BC = 696$, and a right angle, $\angle ABC$. Let $r$ and $s$ denote the inradius and semiperimeter of $ABC$ respectively. Find the perimeter of the triangle ABC which minimizes $\frac{s}{r}$.

Solution

First, label the other leg $x$ and the hypotenuse $y$. To minimize

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