Difference between revisions of "2006 iTest Problems/Problem U7"
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| − | 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 <math>\frac{s}{r}</math>, <math>r</math> must be minimized and <math>s</math> must be maximized. Through logic, it becomes clear that the triangle must be as close to equilateral as possible to maximize <math>r</math> and minimize <math>s</math>(Think about stretching one vertice of an equilateral triangle. The perimeter increases faster than the inradius). | ||
Revision as of 20:42, 17 November 2019
Problem
Triangle 
has integer side lengths, including 
, and a right angle, 
. Let 
and 
denote the inradius and semiperimeter of respectively. Find the perimeter of the triangle ABC which minimizes 
.
Solution
First, label the other leg 
and the hypotenuse 
. To minimize , must be minimized and must be maximized. Through logic, it becomes clear that the triangle must be as close to equilateral as possible to maximize and minimize (Think about stretching one vertice of an equilateral triangle. The perimeter increases faster than the inradius).
