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 <math>\frac{s}{r}</math>, <math>r</math> must be | + | 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 maximize and <math>s</math> must be minimized. 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). |
From the Pythagorean theorem, <math>y^2-x^2=696^2</math>, applying difference of squares yields <math>(y-x)(y+x)=696^2</math>. Since the question states <math>x</math> and <math>y</math> must be integers, we can find possible values of <math>x</math> and <math>y</math> by finding the prime factorization of <math>696^2</math>, which is <math>2^6 \cdot 3 \cdot 29</math>. The two values of <math>x</math> and <math>y</math> that are closest to each other are the values that satisfy | From the Pythagorean theorem, <math>y^2-x^2=696^2</math>, applying difference of squares yields <math>(y-x)(y+x)=696^2</math>. Since the question states <math>x</math> and <math>y</math> must be integers, we can find possible values of <math>x</math> and <math>y</math> by finding the prime factorization of <math>696^2</math>, which is <math>2^6 \cdot 3 \cdot 29</math>. The two values of <math>x</math> and <math>y</math> that are closest to each other are the values that satisfy |
Revision as of 22:32, 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 maximize and
must be minimized. 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).
From the Pythagorean theorem, , applying difference of squares yields
. Since the question states
and
must be integers, we can find possible values of
and
by finding the prime factorization of
, which is
. The two values of
and
that are closest to each other are the values that satisfy
, and
. Solving the system yields
and
. Thus, the perimeter is
Please check my solution as there may be errors.