Question 544544
{{{drawing( 275, 200, -1, 10, -1, 7,
  rectangle( 0, 0, 0.5, 0.5 ), triangle( 0, 0, 9, 0, 0, 6 ),
  locate( 4, 0, 9 ), locate( -0.5, 3.5, 6 ), locate( 4.5, 4, x )
  )}}} {{{x^2=6^2+9^2=36+81=117}}}--> {{{x=sqrt(117)=sqrt(9*13)=sqrt(9)*sqrt(13)=3sqrt(13)}}}
{{{drawing( 225, 225, -1, 8, -1, 8,
  rectangle( 0, 0, 0.5, 0.5 ), triangle( 0, 0, 7, 0, 0, 7 ),
  locate( 3, 0, 7 ), locate( -0.5, 3.5, 7 ), locate( 3.5, 4, x )
  )}}} {{{x^2=7^2+7^2=49+49=98}}}--> {{{x=sqrt(98)=sqrt(49*2)=sqrt(49)*sqrt(2)=7sqrt(2)}}}
The area of rectangle with sides measuring {{{3sqrt(13)}}} and {{{7sqrt(2)}}} is
{{{(3sqrt(13))*(7sqrt(2))=3*7*sqrt(13)*sqrt(2)=21sqrt(26)}}}
The ratio of the hypotenuse lengths found above is
{{{(3sqrt(13))/(7sqrt(2))=((3sqrt(13))/(7sqrt(2)))*(sqrt(2)/sqrt(2))
=(3sqrt(13)sqrt(2))/(7sqrt(2)sqrt(2))=3sqrt(26)/(7*2)=3sqrt(26)/14}}}
I multiplied times the factor {{{1=sqrt(2)/sqrt(2)}}} to rationalize (meaning getting rid of square roots in the denominator).