Voronoi diagrams: exam review Exam review: second paper

Three libraries are at \(A(3, 9), B(1, 2), C(10, 6)\), where the units are kilometres. Each home uses its nearest library, and the Voronoi diagram shows the area served by each library. A home is at \(H(6, 4)\). [6]

1. (a) Find the distance from H to each library. [3]
2. (b) Write down the library that is nearest to H. [1]
3. (c) A second home G is the same distance from library B and from library C, and these are its two nearest libraries. Say where G is on the Voronoi diagram. [1]
4. (d) Give one thing that the Voronoi model leaves out. [1]

A Voronoi diagram has the sites \(P(-3, -1)\) and \(Q(5, 5)\). The line segment RS has equation \(5x+4y=22\) and midpoint \((2, 3)\). [6]

1. (a) Find the midpoint of PQ. [1]
2. (b) Find the equation of the perpendicular bisector of PQ, in the form \(y=mx+c\). [3]
3. (c) Find the equation of the perpendicular bisector of RS, in the form \(px+qy=r\), where p, q and r are integers. [2]

A lake has three fishing huts, A, B and C, where the units are hundreds of metres. The diagram shows the Voronoi diagram for the three sites \(A(3, 1), B(9, 5), C(5, 8)\). The point P\((8, 3)\) is marked. [6]

1. (a) Write down the number of edges in the diagram, and the name of the point where they meet. [2]
2. (b) Find the equation of the edge between A and B, in the form \(px+qy=r\). [3]
3. (c) Write down the site that is nearest to P, and give a reason. [1]

A city measures air pollution with monitors in a park. The diagram shows the Voronoi diagram for the six monitors A to F, with the pollution level at each site. One unit on each axis is 100 m. The points \(X(2, 2)\) and \(Y(7, 3)\) are marked. [7]

1. (a) Use nearest neighbour interpolation to estimate the pollution level at X. [1]
2. (b) The pollution level at X was later measured as 17 units. Find the percentage error in the estimate. [2]
3. (c) The point Y is close to an edge. Find the distance from Y to F and to C, and hence estimate the pollution level at Y. [3]
4. (d) Explain why the estimate at Y is less reliable than the estimate at X. [1]

Two radio masts are at \(A(9, 2)\) and \(B\), where the units are kilometres. The perpendicular bisector of AB has equation \(3x-2y=10\). [8]

1. (a) Find the equation of the line AB, in the form \(y=mx+c\). [2]
2. (b) Use your calculator to find the coordinates of the midpoint M of AB. [2]
3. (c) Hence find the coordinates of B. [1]
4. (d) A straight railway has equation \(y=x\). A station on the railway is the same distance from the two masts. Find the coordinates of the station. [2]
5. (e) Find the distance from the station to A, in kilometres, to 3 significant figures. [1]

Four bus stops are at \(A(5, 2), B(7, 4), C(4, 4), D(3, 5)\), where one unit is 100 m. The diagram shows the Voronoi diagram for the four stops. A new stop E is built at \(E(5, 3)\), and the Voronoi diagram is redrawn. [10]

1. (a) Find the equation of the perpendicular bisector of E and C, in the form \(y=mx+c\). [3]
2. (b) The perpendicular bisector of E and B has equation \(4x+2y=31\). Use your calculator to find the coordinates of the vertex where this bisector meets the bisector from part (a). [2]
3. (c) The old diagram has a vertex at \(V(\frac{11}{2}, \frac{7}{2})\). Show that V is not a vertex of the new diagram. [3]
4. (d) Write down the stops whose cells share an edge with E, and the number of edges of the cell of E. [2]

A national park has five visitor centres at \(A(5, 5), B(8, 5), C(8, 1), D(7, 3), E(3, 3)\), where one unit is 1 km. The diagram shows their Voronoi diagram in a rectangular region \(0\le x\le 9\), \(0\le y\le 7\). The vertices inside the region are P\((5, \frac{3}{4})\), Q\((5, 3)\), R\((\frac{13}{2}, \frac{9}{2})\). A waste site is to be placed anywhere in the region, including its boundary, as far as possible from every centre. [10]

1. (a) Find the distance from each of P, Q and R to its nearest centre. Give your answers to 3 significant figures. [3]
2. (b) Write down which of P, Q and R is farthest from its nearest centre. [1]
3. (c) Find the distance from each corner of the region to its nearest centre, to 3 significant figures. [3]
4. (d) Hence write down where the waste site should be placed, and its distance from the nearest centre. [2]
5. (e) A rule says the waste site must be at least 5.5 units from every centre. Can the rule be met? Give a reason. [1]

Three farms are at \(A(9, 3), B(11, 9), C(5, 11)\), where one unit is 20 km. A waste site is to be placed as far as possible from all three farms. Assume that it goes at the vertex V of their Voronoi diagram. [10]

1. (a) Find the equation of the perpendicular bisector of A and B, in the form \(y=mx+c\). [3]
2. (b) Find the equation of the perpendicular bisector of B and C, in the form \(y=mx+c\). [2]
3. (c) Use your calculator to find the coordinates of V. [2]
4. (d) Show that V is the same distance from A, from B and from C. [2]
5. (e) Find the distance from the waste site to the nearest farm, in kilometres, to 3 significant figures. [1]

A lake is a rectangle 10 km by 8 km. Four buoys are at \(A(5, 5), B(7, 4), C(5, 2)\) and \(E(6, 4)\), where one unit is 1 km. The diagram shows the Voronoi diagram for the four buoys. The cell of E is a triangle with vertices \(P(\frac{13}{2}, \frac{11}{2})\), \(Q(\frac{13}{2}, \frac{5}{2})\) and \(R(\frac{9}{2}, \frac{7}{2})\). [10]

1. (a) Find the area of the cell of E. [2]
2. (b) Find the percentage of the lake that is nearer to E than to any other buoy, to 3 significant figures. [2]
3. (c) Buoy E is now removed. Find the equation of the perpendicular bisector of A and B, in the form \(y=mx+c\). [3]
4. (d) The perpendicular bisector of B and C is \(y=-x+9\). Use your calculator to find the coordinates of the new vertex where the cells of A, B and C meet. [2]
5. (e) Write down the number of vertices of the old diagram that are removed. [1]