Question 335286
Square both sides.
{{{x^2+y^2=(x+y)^2}}}
{{{x^2+y^2=(x+y)(x+y)}}}
{{{x^2+y^2=x^2+xy+yx+y^2}}}
{{{x^2+y^2=x^2+2xy+y^2}}}
{{{2xy=0}}}
So the above equality only holds if either x or y or both are zero. 
Otherwise you're missing the 2xy term.
{{{sqrt(x^2+2xy+y^2)=sqrt((x+y)^2)=x+y}}}