1. **Problem statement:** Find the slope of a line passing through the origin and the midpoint of the segment joining points $P(0,-4)$ and $B(8,0)$. 2. **Formula for midpoint:** The midpoint $M$ of segment joining points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ 3. **Calculate midpoint $M$:** $$M = \left(\frac{0+8}{2}, \frac{-4+0}{2}\right) = (4, -2)$$ 4. **Formula for slope $m$ of line through points $(x_1,y_1)$ and $(x_2,y_2)$:** $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 5. **Calculate slope of line through origin $(0,0)$ and midpoint $M(4,-2)$:** $$m = \frac{-2 - 0}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}$$ **Final answer:** The slope of the line is $-\frac{1}{2}$.