Answer: \(y-1=\frac34(x+2)\).

The midpoint of the line segment is

\(\left(\frac{4+(-8)}2,\frac{-7+9}2\right)=(-2,1).\)

The gradient of the line joining the two points is

\(\frac{9-(-7)}{-8-4}=\frac{16}{-12}=-\frac43.\)

The gradient of the perpendicular bisector is therefore the negative reciprocal:

\(\frac34.\)

Using the point \((-2,1)\), the perpendicular bisector is

\(\boxed{y-1=\frac34(x+2)}.\)