Equation of a perpendicular bisector Problem set A

A Voronoi diagram has the three sites \(P(0, 2)\), \(Q(6, 8)\), \(R(10, 0)\). For each pair of sites, write down the midpoint and find the equation of the perpendicular bisector, in the form \(y=mx+c\).

1. (a) Sites P and Q.
2. (b) Sites P and R.
3. (c) Sites Q and R.

Find the equation of the perpendicular bisector of the line segment joining \((-3, 6)\) and \((5, 0)\), in the form \(y=mx+c\).

The line segment AB has equation \(3x-2y=-1\) and midpoint \((3, 5)\). Find the equation of the perpendicular bisector of AB, in the form \(px+qy=r\), where p, q and r are integers.

Find the equation of the perpendicular bisector of the line segment joining:

1. (a) \((4, 1)\) and \((4, 7)\);
2. (b) \((-1, 5)\) and \((9, 5)\).

Two cafés are sites in a Voronoi diagram. One is at \(S(3, 1)\). The edge between the two cells has equation \(y=5\). Find the coordinates of the other café, T.

Two phone masts are at \(M(2, 9)\) and \(N(8, 3)\), where the units are kilometres. Each house uses the nearer mast. The boundary between the two areas is the perpendicular bisector of MN. A house is at \(H(3, 7)\).

1. (a) Find the equation of the boundary, in the form \(y=mx+c\).
2. (b) Find which mast the house at H uses, by working out the two distances.

Two wells are at \(W(1, 10)\) and \(V(9, 0)\). The line through them has equation \(5x+4y=45\) and its midpoint is \((5, 5)\). Each household uses the nearer well. Find the equation of the boundary between the two areas, in the form \(px+qy=r\).

Two farms are at \(P(9, 5)\) and \(Q(5, 3)\), where the units are kilometres. A straight canal has equation \(y=x+6\). A pump on the canal must be the same distance from the two farms.

1. (a) Find the equation of the perpendicular bisector of PQ, in the form \(y=mx+c\).
2. (b) Find the coordinates of the pump.

The sites \(A(2, 1)\) and \(B(8, 5)\) are in a Voronoi diagram, and K is the point \((3, 6)\).

1. (a) Find the equation of the perpendicular bisector of AB, in the form \(y=mx+c\).
2. (b) Show that K is on the bisector by showing that K is the same distance from A and from B.

\(A(2, 1)\) is a site. The perpendicular bisector of AB has equation \(y=-2x+10\). Find the coordinates of B.