Question 355054
The area of the triangle is {{{A=(1/2)a*b=(1/2)a*6 = 3a}}}.
The perimeter, on the other hand, is {{{P= a+b+c = a+6+c}}}.  Since area = perimeter, then {{{3a = a+6+c}}}, or {{{2a = 6+c}}}, or {{{c = 2a - 6}}}.
By the Pythagorean theorem,
{{{c^2=a^2+b^2}}},
{{{(2a-6)^2 = a^2 + 6^2}}},
{{{4a^2 - 24a +36 = a^2 + 36}}},
{{{3a^2-24a = 0}}},
{{{a^2 - 8a = 0}}}, or
{{{a*(a-8) = 0}}}.  Therefore {{{a = 0}}} or {{{a = 8}}}.  Discard the first value, so {{{a = 8}}}.  The hypotenuse is then {{{c = 2*8-6 = 10}}}.