[tex]\text{Let}\ k:y=m_1x+b_1\ \text{and}\ l:y=m_2x+b_2\\\\l\perp k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}\\\\\text{We have the points B(6, 9) and C(8, -7)}.\\\\\text{The formula of a slope:}\\\\m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\text{substitute:}\\\\m_1=\dfrac{-7-9}{8-6}=\dfrac{-16}{2}=-8\\\\\text{therefore}\ m_2=-\dfrac{1}8}=-\dfrac{1}{8}\\\\\text{The formula of a midpoint:}\\\\\left(\dfrac{x_1+x_2}{2};\ \dfrac{y_1+y_2}{2}\right)\\\\\text{substitute the coordinates of the points B and C:}[/tex]
[tex]x=\dfrac{6+8}{2}=\dfrac{14}{2}=7\\\\y=\dfrac{9+(-7)}{2}=\dfrac{2}{2}=1\\\\\text{midpoint}\ (7,\ -1)\\\\\text{The point-slope form:}\\\\y-y_1=m(x-x_1)\\\\\text{Substitute}\ m=-\dfrac{1}{8},\ x_1=7\ \text{and}\ y_1=1:\\\\\boxed{y-1=-\dfrac{1}{8}(x-7)}[/tex]
