Find the co-ordinates of the mid point of the line segment joining the points A(–3,2) and B(7,8).

A triangle whose vertices are the points A(1,–2), B(3,1) and C(–2,3).

If AD is drawn perpendicular to BC then its slope is \$5/2\$.
Equation of AD is

\begin{align} && y + 2 &= \frac{5}{2} (x – 1) \\ \Rightarrow && 5x – 2y &= 9 \tag{1} \\ \end{align}

Solving equation (1) and (2).

The orthocentre is the point of intersection of altitudes AD and BE.

\begin{align} 15x – 6y = 27& \\ \underline{+15x – 25y = 20}& \\ 19y = 7& \\ \\ y = \frac{7}{19} \\ \end{align}

Substitute \$ y = \frac{7}{19}\$ in (1), we get

\begin{align} \require{cancel} 5x - 2(\frac{7}{19} ) &= 9 \\ x &= \frac{37}{19} \\ \end{align}

Answer

The ortho centre is \$( \frac{37}{19}, \frac{7}{19})\$.