Question 1073186
x, longest side
y, shortest side
r, the diagonal path


Total path actually used, {{{r=x+y-0.8y}}}
{{{r=x+0.2y}}}
{{{r=x+y/5}}}
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Question means, to find the ratio {{{y/x}}}.




r is also hypotenuse of the right triangle of sides x and y.
{{{r^2=x^2+y^2}}}



Also from formula for r, {{{r^2=(x+y/5)^2=x^2+2xy/5+y^2/25}}}


The two expressions for r^2 are equal.
{{{x^2+y^2=x^2+2xy/5+y^2/25}}}
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Algebra steps
{{{y^2=2xy/5+y^2/25}}}


{{{25y^2=10xy+y^2}}}


Divide both sides by {{{y^2}}}.

{{{25y^2/y^2=10xy/y^2+y^2/y^2}}}

{{{25=10x/y+1}}}

{{{24=10x/y}}}

{{{24/10=x/y}}}

{{{y/x=10/24}}}

{{{highlight(y/x=5/12)}}}, ratio of shortest side to longest side