Sites, cells, edges and vertices Problem set A

The diagram shows the Voronoi diagram for the sites \(A(11, 2)\), \(B(5, 2)\), \(C(3, 10)\).

1. (a) Write down the number of cells and the number of edges.
2. (b) Write down the coordinates of the vertex.
3. (c) What is the point where the edges meet called?

The sites \(A(4, 6)\), \(B(4, 2)\), \(C(3, 11)\), \(D(8, 6)\) are plotted on the grid. The sites A and D have an edge between two vertices.

1. (a) Find the equation of the edge between A and D, in the form \(px+qy=r\).
2. (b) Draw the Voronoi diagram on a copy of the grid. Write down the coordinates of the two vertices.

The diagram shows the Voronoi diagram for the sites \(A(8, 7)\), \(B(10, 10)\), \(C(5, 10)\), \(D(1, 7)\). The point \(P(11, 8)\) is marked.

1. (a) Which site is nearest to P?
2. (b) Show that this is true by working out the distance from P to this site and to the next nearest site.

The sites \(A(6, 3)\), \(B(1, 6)\), \(C(11, 6)\), \(D(1, 2)\) are plotted on the grid.

1. (a) Do the cells of A and B share an edge? Use the midpoint of AB to explain.
2. (b) Do the cells of C and D share an edge? Use the midpoint of CD to explain.

Five fire stations are at \(A(4, 6)\), \(B(10, 8)\), \(C(6, 2)\), \(D(10, 4)\), \(E(7, 5)\), where the units are kilometres. The diagram shows three of the edges of the Voronoi diagram, dashed: the edges between E and A, B, C. The fourth edge of the cell containing E is the edge between E and D. A house is at \(Q(7, 4)\).

1. (a) Find the equation of the line that forms the final edge of the cell containing E, in the form \(px+qy=r\).
2. (b) Sketch the cell containing E and write down the coordinates of its vertices.
3. (c) The house at \(Q(7, 4)\) lies in the cell containing E. What does this tell you about the station nearest to the house?

Phone masts are at \(A(3, 1)\), \(B(11, 1)\), \(C(11, 6)\), where the units are kilometres. A fourth mast D is built so that ABCD is a rectangle.

1. (a) Write down the coordinates of the fourth mast D.
2. (b) Find the coordinates of the vertex where four edges meet, and show that it is the same distance from all four masts.
3. (c) Explain why the cells of A and C do not share an edge.

A national park is a rectangle 12 units by 8 units on a map, where one unit is 2 km. Four water points are at \(A(6, 1)\), \(B(2, 7)\), \(C(8, 5)\) and \(E(6, 3)\). The cell of E is a triangle with vertices \((1, 2)\), \((9, 2)\) and \((5, 6)\).

1. (a) Find the area of the cell of E, in square units.
2. (b) Find the area of the cell of E, in km².
3. (c) Find the percentage of the park that is nearer to E than to any other water point.

Three schools are at \(A(4, 1)\), \(B(4, 9)\), \(C(9, 4)\), where the units are kilometres. The edges of the Voronoi diagram between them meet at a vertex V.

1. (a) Find the equations of the edges between A and B and between A and C, in the form \(px+qy=r\).
2. (b) Use your calculator to find the coordinates of the vertex V where these two edges meet.
3. (c) Show that V is the same distance from A, B and C.

Four wind turbines are at \(A(11, 8)\), \(B(5, 1)\), \(C(2, 6)\), \(D(6, 5)\), where the units are hundreds of metres. D is inside the triangle formed by A, B and C. Every point is assigned to its nearest turbine.

1. (a) Find the coordinates of the vertices of the cell containing D.
2. (b) How many edges does D's cell have? What does this tell you about where D is?