3.6 Voronoi diagrams questions for IB Maths AI

36 questions on Voronoi diagrams, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 3 Geometry and trigonometry
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	19	0	19
Paper 2	16	0	16
Paper 3HL only	0	1	1
Every paper	35	1	36

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00113Paper 1 · 6 marks

A telecommunications company has three mobile phone masts at A $(3, 2)$ , B $(3, 10)$ and C $(11, 10)$ , where distances are measured in kilometres.

(a)
Write down the equation of the perpendicular bisector of [AB].
2 marks
(b)
Write down the equation of the perpendicular bisector of [BC].
2 marks
(c)
A maintenance depot is to be built at the point that is the same distance from all three masts. Find the distance from the depot to each mast.
2 marks

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Question 2 · AI-00227Paper 2 · 15 marks

A phone network has towers T₁(3, 4) and T₂(9, $b$ ) on a coordinate grid where one unit represents one kilometre. Every point on the boundary line L between the coverage areas of T₁ and T₂ is equidistant from the two towers. L has equation $3 x + y = 23$ .

(a)
Write down the gradient of L.
1 mark
(b)
Hence write down the gradient of [T₁T₂].
1 mark
(c)
Find the value of $b$ .
2 marks
(d)
Verify that the midpoint of [T₁T₂] lies on L.
2 marks
(e)
A third tower is built at T₃(7, 12). Find the equation of the perpendicular bisector of [T₂T₃].
4 marks
(f)
Find the coordinates of the point V that is equidistant from all three towers.
2 marks
(g)
Find the distance from V to each tower.
2 marks
(h)
The engineers say that V is where the signal from the network is likely to be weakest. Give a reason why they might say this.
1 mark

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Question 3 · AI-01800Paper 3 · HL only · 27 marks

The purpose of this question is to use Voronoi diagrams to help a city council decide where to build a new public library.

Mehmet is a planner for a city whose boundary is modelled by the rectangle $0 \leq x \leq 12$ , $0 \leq y \leq 10$ , where one unit represents 1 km. The city has five libraries, at $A (1, 3)$ , $B (7, 1)$ , $C (11, 5)$ , $D (7, 9)$ and $E (2, 9)$ . Residents are assumed to use the library nearest to their home.

The following diagram shows the libraries and part of the Voronoi diagram for the five libraries. The edge between the cells of $A$ and $B$ is missing.

Grid with 0 ≤ x ≤ 12, 0 ≤ y ≤ 10. Libraries A(1, 3), B(7, 1), C(11, 5), D(7, 9), E(2, 9). Voronoi edges drawn: (0, 6.25) to (4.5, 5.5); (4.5, 5.5) to (4.5, 10); (4.5, 5.5) to (5, 5); (5, 5) to (7, 5); (7, 5) to (12, 0); (7, 5) to (12, 10). The edge between A and B is missing.

(a)
Find the equation of the perpendicular bisector of $[A B]$ . Give your answer in the form $y = m x + c$ .
4 marks
(b)
Show that the point $(5, 5)$ is equidistant from $A$ , $B$ and $D$ .
3 marks
(c)
Hence find the coordinates of the end points of the missing edge.
2 marks
(d)
The council wants to build a new library at the point in the city that is furthest from its nearest existing library. A councillor claims that this point must be a vertex of the Voronoi diagram.
The vertices of the Voronoi diagram are $(4.5, 5.5)$ , $(5, 5)$ and $(7, 5)$ .
For each vertex, find the distance to its nearest library. Hence state which vertex the councillor would choose.
3 marks
(e)
By considering the corner $(12, 0)$ of the city, show that the councillor's claim is incorrect.
2 marks
(f)
Suggest one reason why the council might still prefer to build the new library at $(5, 5)$ .
1 mark
(g)
The council builds library $F$ at $(5, 5)$ .
(i) Write down the equation of the perpendicular bisector of $[F C]$ .
(ii) Find the equation of the perpendicular bisector of $[F B]$ .
3 marks
(h)
In the new Voronoi diagram, the cell of $F$ is a pentagon with vertices $(4, 2)$ , $(8, 4)$ , $(8, 6)$ , $(4.5, 7.75)$ and $G$ . The point $G$ is the intersection of the perpendicular bisectors of $[F A]$ and $[F E]$ , which have equations $y = - 2 x + 10$ and $3 x - 4 y = - 17.5$ .
Find the coordinates of $G$ .
2 marks
(i)
The area of a polygon with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ , listed in order around the polygon, is
$\frac{1}{2} ∣ (x_{1} y_{2} - x_{2} y_{1}) + (x_{2} y_{3} - x_{3} y_{2}) + \dots + (x_{n} y_{1} - x_{1} y_{n}) ∣ .$
Use this formula to find the area of the cell of $F$ .
4 marks
(j)
The population density of the city is approximately 1800 people per km².
Estimate the number of residents who will use library $F$ .
2 marks
(k)
State one reason why the actual number of people using library $F$ may be different from this estimate.
1 mark

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Question 4 · AI-00880Paper 1 · 5 marks

The following Voronoi diagram shows the locations of four supermarkets, A, B, C and D, in a town. One unit on each axis represents one kilometre. Kalinda's house is located at K(7, 3).

Voronoi diagram on a grid, x from 0 to 10 and y from 0 to 9, for supermarkets A(2, 6), B(7, 7), C(3, 2) and D(8, 2). The cell containing B is shaded. Kalinda's house K is at (7, 3).

(a)
Write down which supermarket is closest to Kalinda's house.
1 mark
(b)
Write down the number of vertices in the Voronoi diagram.
1 mark
(c)
Explain, in context, what the shaded cell represents.
1 mark
(d)
Find the distance from Kalinda's house to the nearest supermarket.
2 marks

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Question 5 · AI-00266Paper 2 · 15 marks

The town of Aldmoor has three fire stations, at A $(2, 2)$ , B $(10, 6)$ and C $(4, 10)$ , where distances are measured in kilometres. The following diagram shows the positions of the three stations and the perpendicular bisector of [BC], which has equation $y = 1.5 x - 2.5$ .

perpendicular bisector of [BC]

In the event of a fire, the fire station closest to the fire responds.

(a)
Write down
(i) the coordinates of the midpoint of [AB];
(ii) the gradient of [AB].
2 marks
(b)
Show that the perpendicular bisector of [AB] has equation $y = - 2 x + 16$ .
2 marks
(c)
Find the coordinates of V, the point where the perpendicular bisectors of [AB] and [BC] meet.
2 marks
(d)
Find the equation of the perpendicular bisector of [AC], and hence verify that V lies on it.
3 marks
(e)
Find the distance from V to each of the three fire stations.
2 marks
(f)
A fire breaks out at a warehouse at H $(6, 2)$ . Determine which fire station responds.
2 marks
(g)
The town council requires every point in Aldmoor to be within $5$ km of its nearest fire station. A planner says that V is the point in the triangle ABC furthest from its nearest station. Hence determine whether every point inside the triangle ABC is within $5$ km of a fire station.
2 marks

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Question 6 · AI-00881Paper 1 · 6 marks

Three weather stations are located at A(2.5, 1), B(5.5, 5) and C(9, 1). The lines $L_{1}$ , $L_{2}$ and $L_{3}$ shown on the following diagram are the perpendicular bisectors of [AB], [AC] and [BC] respectively.

Grid with x from 0 to 10 and y from 0 to 8 showing sites A(2.5, 1), B(5.5, 5) and C(9, 1) and three lines: L1 through (0, 6) and (8, 0)

(a)
Write down the equation of $L_{1}$ .
2 marks
(b)
The equation of $L_{2}$ is $x = 5.75$ .
Find the coordinates of the point V where $L_{1}$ and $L_{2}$ intersect. Give your answer correct to four significant figures.
2 marks
(c)
Find the distance from V to B.
2 marks

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