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IB Mathematics: Applications and Interpretation HL · 12.8: Voronoi Diagrams · 12.8.2

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Equation of a perpendicular bisector Problem set B

Find the midpoint first. Give answers in the form asked for. A calculator can be used to check, not to replace the working.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A Voronoi diagram has the three sites \(D(1, 7)\), \(E(9, 3)\), \(F(5, 11)\). 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 D and E.
    2. (b) Sites D and F.
    3. (c) Sites E and F.

  2. Find the equation of the perpendicular bisector of the line segment joining \((-6, -1)\) and \((2, 5)\), in the form \(y=mx+c\).

  3. The line segment AB has equation \(x+4y=12\) and midpoint \((4, 2)\). Find the equation of the perpendicular bisector of AB, in the form \(px+qy=r\), where p, q and r are integers.

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

    1. (a) \((2, -3)\) and \((2, 5)\);
    2. (b) \((-4, 6)\) and \((2, 6)\).

  5. Two kiosks are sites in a Voronoi diagram. One is at \(U(-4, 3)\). The edge between the two cells has equation \(y=-1\). Find the coordinates of the other kiosk, V.

In context

  1. Two schools are at \(S(1, 2)\) and \(T(7, 10)\), where the units are hundreds of metres. Each pupil goes to the nearer school. The boundary between the two catchment areas is the perpendicular bisector of ST. A pupil lives at \(H(5, 7)\).

    1. (a) Find the equation of the boundary, in the form \(y=mx+c\).
    2. (b) Find which school the pupil goes to, by working out the two distances.

  2. Two lifeguard towers are at \(T(2, 1)\) and \(U(6, 9)\) on a map. The line through them has equation \(2x-y=3\) and its midpoint is \((4, 5)\). Each swimmer is helped by the nearer tower. Find the equation of the boundary between the two areas, in the form \(px+qy=r\).

  3. Two fire stations are at \(R(11, 5)\) and \(T(1, 7)\), where the units are kilometres. A straight cycle path has equation \(x-y=4\). A drinking fountain on the path must be the same distance from the two stations.

    12345678910111212345678910xyRTF
    1. (a) Find the equation of the perpendicular bisector of RT, in the form \(y=mx+c\).
    2. (b) Find the coordinates of the fountain.

Stretch

  1. The sites \(C(1, 6)\) and \(D(7, 2)\) are in a Voronoi diagram, and K is the point \((0, -2)\).

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

  2. \(C(3, 2)\) is a site. The perpendicular bisector of CD has equation \(y=-3x+21\). Find the coordinates of D.

jmaths.xyz · AI HL · 12.8.2 Equation of a perpendicular bisector