Question 785524
{{{a}}}= length of one leg, in meters
{{{b}}}= length of the other leg, in meters
{{{h}}}= hypotenus of the triangle, in meters
{{{ab/2=25}}}= area of the triangle, in square meters
{{{p=a+b+h}}}= perimeter of the triangle, in square meters
 
{{{ab/2=25}}}-->{{{ab=25*2}}}-->{{{ab=50}}}
{{{p=a+b+h}}}-->{{{p-h=a+b}}}
According to the Pythagorean theorem:
{{{h^2=a^2+b^2}}}
{{{p^2=(a+b+h)^2=a^2+b^2+h^2+2ab+2ah+2bh}}}
{{{p^2-h^2=h^2+2ab+2ah+2bh}}}
{{{p^2-h^2=h^2+2*50+2ah+2bh}}} (substituting {{{ab=50}}})
{{{p^2=2h^2+100+2ah+2bh}}}
{{{p^2=2h^2+100+2h(a+b)}}}
{{{p^2=2h^2+100+2h(p-h)}}} (substituting {{{p-h=a+b}}})
{{{p^2=2h^2+100+2hp-2h^2}}}
{{{p^2=100+2hp}}}
{{{p^2-100=2hp}}}
{{{highlight(h=(p^2-100)/2p)}}}
or something equivalent, like
{{{highlight(h=p/2-50/p)}}} or {{{highlight(h=(p+10)(p-10)/2p)}}}
 
EXTRA:
A more general expression, bases on perimeter and area {{{A}}} would be
{{{highlight(h=(p^2-4A)/2p)}}}