a) Compute x+y and $\sqrt{x^2+y^2}$ when x=5 and y=12

$$\\ a) \text{ Compute x+y and} \sqrt{x^2+y^2} \text{ when } x=5 \text{ and } y=12\\\\ b) \text{ When is } \left[x+y=\sqrt{x^2+y^2}~?\right]\\ \text{ When is } \left[x+y\neq\sqrt{x^2+y^2}~?\right]$$

a)

$$\small{\text{$ \begin{array}{rcl} x+y &=& 5 + 12\\ x+y &=& 17 \\\\ \sqrt{x^2+y^2}&=& \sqrt{5^2+12^2}\\ \sqrt{x^2+y^2}&=& \sqrt{25+144}\\ \sqrt{x^2+y^2}&=& \sqrt{169}\\ \sqrt{x^2+y^2}&=& \sqrt{13^2}\\ \sqrt{x^2+y^2}&=& 13\\ \end{array} $}}$$

b)

$$x+y \stackrel{\textcolor[rgb]{1,0,0}{?}} = \sqrt{x^2+y^2} \qquad \text{ squaring } \\\\ (x+y)^2 \stackrel{\textcolor[rgb]{1,0,0}{?}} = x^2+y^2\\\\ x^2+2xy+y^2 \stackrel{\textcolor[rgb]{1,0,0}{?}}= x^2+y^2 \\\\ \boxed{~~ x^2+\underbrace{\textcolor[rgb]{1,0,0}{2xy}}_{=0}+y^2 = x^2+y^2 ~~}\\\\ \boxed{~~ x^2+\underbrace{\textcolor[rgb]{1,0,0}{2xy}}_{ \neq0}+y^2 \neq x^2+y^2 ~~}$$