Find the orthocentre of the triangle with the vertices (–5, –7), (13, 2) & (–5, 6)

Let A(–5, –7), B(13, 2) and C(–5, 6) be the given vertices

$\text{ Let }\overline{\mathrm{AD}}\perp \overline{\mathrm{BC}}$
$\text{ Slope of }\overline{\mathrm{BC}}=-2/9$

$\text{ Let }\overline{\mathrm{AD}}\perp \overline{\mathrm{BC}}$ then

slope of $AD$=$\frac{9}{2}$
$\text{ Equation of }\overline{\mathrm{AD}}\text{ is }\mathrm{y}-(-7)=\frac{9}{2}(\mathrm{x}-(-5)$
$\Rightarrow 9\mathrm{x}-2\mathrm{y}+31=0 ....\left(1\right)$
$\text{ And }\overline{\mathrm{BE}}\perp \overline{\mathrm{AC}}\text{, slope of }\overline{\mathrm{AC}}=\frac{6+7}{-5+5}=\frac{13}{0}$
$\text{ Equation of }\overline{\mathrm{BE}}\text{ is }\mathrm{y}-2=\frac{0}{13}(\mathrm{x}-13)\Rightarrow \mathrm{y}=2 ....\left(2\right)$
Solve (1) & (2) x = –3, y = 2
orthocentre of the triangle  is (–3, 2)