$BC=\sqrt{{{(2-4)}^{2}}+{{(4-(-1))}^{2}}}=\sqrt{4+25}=\sqrt{29}$ $BC:{{m}_{BC}}=\frac{4+1}{2-4}=-\frac{5}{2}$ $BC:y-4=-\frac{5}{2}(x-2)\Rightarrow y-4=-\frac{5}{2}x=5$ $\Rightarrow \frac{5}{2}x+y-9=0\Rightarrow 5x+2y-18=0$ ارتفاع AH برابر است با فاصلهٔ نقطهٔ $A(-1,2)$ از خط 5x+2y-18=0. $AH=\frac{\left| -5+4-18 \right|}{\sqrt{25+4}}=\frac{\left| -19 \right|}{\sqrt{29}}=\frac{19}{\sqrt{29}}$ $\Rightarrow S\overset{\vartriangle }{\mathop{ABC}}\,=\frac{1}{2}\times \frac{19}{\sqrt{29}}\times \sqrt{29}=\frac{19}{2}$