3.6 Voronoi diagramsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A Voronoi diagram is drawn for three schools, which are the sites $A(2,1)$, $B(10,1)$ and $C(6,9)$. Coordinates are in km.

   (a)

   Find the equation of the edge between the cells of $A$ and $B$.

   [1 mark]

   • A$y=1$
   • B$x=4$
   • C$x=6$
   • D$y=5$

   (b)

   The three edges of the diagram meet at the vertex $(6,4)$. Find the distance from this vertex to school $C$.

   [1 mark]

   • A$3$ km
   • B$4$ km
   • C$9$ km
   • D$5$ km

   (c)

   Find the equation of the edge between the cells of $A$ and $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Rainfall is recorded at three weather stations: $P(1,2)$ recorded 120 mm, $Q(7,2)$ recorded 80 mm and $R(4,8)$ recorded 60 mm. Coordinates are in km. The rainfall at any other point is estimated by nearest neighbour interpolation, using the Voronoi diagram of the three stations.

   (a)

   Estimate the rainfall at the point $X(5,3)$.

   [1 mark]

   • A$80$ mm
   • B$120$ mm
   • C$60$ mm
   • D$86.7$ mm

   (b)

   Which of these points is estimated to have a rainfall of 120 mm?

   [1 mark]

   • A$(5,5)$
   • B$(2,3)$
   • C$(4,7)$
   • D$(6,1)$

   (c)

   A town $T$ is at $(4,6)$. Find the distance from $T$ to each station, and hence estimate the rainfall at $T$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company plans to build a waste facility as far as possible from its nearest village, at a vertex of the Voronoi diagram of three villages $V_1(0,0)$, $V_2(8,0)$ and $V_3(2,6)$. Coordinates are in km. In the Voronoi diagram, the edge between the cells of $V_1$ and $V_2$ is $x=4$, and the edge between the cells of $V_2$ and $V_3$ is $y=x-2$.

   (a)

   Find the equation of the edge between the cells of $V_1$ and $V_3$, giving your answer in the form $x+3y=d$.

   [3 marks]

   (b)

   Find the coordinates of the vertex where the three edges meet, and hence find the distance from the proposed facility to the nearest village.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A nature reserve is the rectangle $0\le x\le12$, $0\le y\le8$, with coordinates in km. Two ranger stations $R_1(3,3)$ and $R_2(9,3)$ are sites of a Voronoi diagram, with the edge $x=6$ between their cells. A third station $R_3(6,7)$ is then added.

   (a)

   (i) Find the equation of the edge between the cells of $R_1$ and $R_3$.
   (ii) Find the coordinates of the vertex where this edge meets the edge $x=6$.
   (iii) Write down the station nearest to the point $(6,6)$.

   [6 marks]

   (b)

   (i) The edge between the cells of $R_1$ and $R_3$ meets the top side of the reserve, $y=8$. Find the $x$-coordinate of this point.
   (ii) By symmetry, the edge between the cells of $R_2$ and $R_3$ meets $y=8$ at $x=11.5$. The cell of $R_3$ inside the reserve is a triangle. Find its area.
   (iii) Find the percentage of the reserve that is nearest to $R_3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions