Voronoi diagrams: exam review Exam review

Three health clinics are at \(A(1, 2), B(9, 4), C(4, 9)\), where the units are kilometres. Each home goes to its nearest clinic, and the Voronoi diagram shows the area served by each clinic. A home is at \(H(6, 5)\). [6]

1. (a) Find the distance from H to each clinic. [3]
2. (b) Write down the clinic that is nearest to H. [1]
3. (c) A second home G is the same distance from clinic A and from clinic B, and these are its two nearest clinics. 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 \(P(-2, 3)\) and \(Q(4, 7)\). The line segment RS has equation \(3x+2y=17\) and midpoint \((3, 4)\). [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 city has three fire stations, A, B and C, where the units are kilometres. The diagram shows the Voronoi diagram for the three sites \(A(2, 2), B(8, 4), C(4, 9)\). The point P\((5, 6)\) 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 farmer has soil-moisture sensors across a field. The diagram shows the Voronoi diagram for the six sensors A to F, with the soil moisture at each site. One unit on each axis is 10 m. The points \(X(3, 5)\) and \(Y(3, 3)\) are marked. [7]

1. (a) Use nearest neighbour interpolation to estimate the soil moisture at X. [1]
2. (b) The soil moisture at X was later measured as 24%. 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 A, and hence estimate the soil moisture at Y. [3]
4. (d) Explain why the estimate at Y is less reliable than the estimate at X. [1]

Two schools are at \(A(2, 9)\) and \(B\), where the units are kilometres. The perpendicular bisector of AB has equation \(3x-2y=1\). [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 road has equation \(y=x+1\). A bus stop on the road is the same distance from the two schools. Find the coordinates of the bus stop. [2]
5. (e) Find the distance from the bus stop to A, in kilometres, to 3 significant figures. [1]

Four weather stations are at \(A(8, 4), B(6, 2), C(4, 4), D(2, 4)\), where one unit is 1 km. The diagram shows the Voronoi diagram for the four stations. A new station E is built at \(E(6, 3)\), 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 A has equation \(4x+2y=35\). 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 \(V(6, 4)\). Show that V is not a vertex of the new diagram. [3]
4. (d) Write down the stations whose cells share an edge with E, and the number of edges of the cell of E. [2]

An island has five villages at \(A(3, 2), B(3, 6), C(8, 3), D(5, 4), E(4, 1)\), 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\((3, 4)\), Q\((\frac{9}{2}, \frac{5}{2})\), R\((6, 2)\). A waste site is to be placed anywhere in the region, including its boundary, as far as possible from every village. [10]

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

Three schools are at \(A(5, 9), B(7, 3), C(5, 5)\), where one unit is 10 km. A waste site is to be placed as far as possible from all three schools. 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 school, in kilometres, to 3 significant figures. [1]

A country park is a rectangle 12 km by 8 km. Four picnic shelters are at \(A(3, 3), B(5, 7), C(7, 7)\) and \(E(5, 6)\), where one unit is 1 km. The diagram shows the Voronoi diagram for the four shelters. The cell of E is a triangle with vertices \(P(1, \frac{13}{2})\), \(Q(6, \frac{13}{2})\) and \(R(\frac{17}{2}, \frac{3}{2})\). [10]

1. (a) Find the area of the cell of E. [2]
2. (b) Find the percentage of the park that is nearer to E than to any other shelter, to 3 significant figures. [2]
3. (c) Shelter 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 \(x=6\). 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]