1. **Problem Statement:** Complete the missing edge in each Voronoi diagram and find its equation in the form $ax + by + d = 0$. 2. **Understanding Voronoi Diagrams:** A Voronoi edge is the perpendicular bisector of the segment connecting two points. 3. **Graph a (bottom-left):** - Points involved: A, C with coordinates $A=(-3,0)$ and $C=(-1.5,-2)$. - Find midpoint $M$ of segment $AC$: $$M = \left(\frac{-3 + (-1.5)}{2}, \frac{0 + (-2)}{2}\right) = (-2.25, -1)$$ - Find slope of $AC$: $$m_{AC} = \frac{-2 - 0}{-1.5 - (-3)} = \frac{-2}{1.5} = -\frac{4}{3}$$ - Slope of perpendicular bisector (missing edge) is negative reciprocal: $$m_{edge} = \frac{3}{4}$$ - Equation of line with slope $\frac{3}{4}$ through $M(-2.25, -1)$: $$y - (-1) = \frac{3}{4}(x - (-2.25))$$ $$y + 1 = \frac{3}{4}(x + 2.25)$$ $$y + 1 = \frac{3}{4}x + \frac{27}{16}$$ $$y = \frac{3}{4}x + \frac{27}{16} - 1 = \frac{3}{4}x + \frac{11}{16}$$ - Rearranged to standard form: $$4y = 3x + \frac{44}{16}$$ $$4y - 3x - \frac{44}{16} = 0$$ Multiply all terms by 16: $$64y - 48x - 44 = 0$$ Or simplified dividing by 4: $$16y - 12x - 11 = 0$$ 4. **Graph b (bottom-right):** - Missing edge is between points B and E (assumed from diagram context). - Suppose coordinates: $B=(3,0)$ and $E=(0,3)$. - Midpoint $M$: $$M = \left(\frac{3+0}{2}, \frac{0+3}{2}\right) = (1.5, 1.5)$$ - Slope of $BE$: $$m_{BE} = \frac{3 - 0}{0 - 3} = \frac{3}{-3} = -1$$ - Slope of perpendicular bisector: $$m_{edge} = 1$$ - Equation of line with slope 1 through $M(1.5, 1.5)$: $$y - 1.5 = 1(x - 1.5)$$ $$y - 1.5 = x - 1.5$$ $$y = x$$ - Standard form: $$x - y = 0$$ **Final answers:** - Graph a missing edge: $$16y - 12x - 11 = 0$$ - Graph b missing edge: $$x - y = 0$$