IB Maths AI SL · Unit 3: Geometry and Trigonometry
Exam-style IB Maths AI SL voronoi diagrams questions with worked solutions. Start with the 4 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
SL AI Paper 2 loves 'which mobile mast is closest to point P?' — a Voronoi cell question. Learn how to construct the perpendicular bisectors by hand and read off the answer. Voronoi diagrams — nearest-neighbour problems overview →
The question bank covers these voronoi diagrams question types (number of questions in brackets):
Boundary $P$–$Q$: $y = 2x - 1$. Boundary $Q$–$R$: $x + y = 14$. Find the Voronoi vertex.
1. System: intersect the two boundary lines.
2. Substitute: $x + (2x - 1) = 14 \implies 3x = 15$.
3. Solve: $x = 5$, $y = 9$.
4. State: $\mathbf{(5, 9)}$.
Voronoi diagram of hospitals $H_1, H_2, H_3$. Accident at point $P$. Which hospital dispatches?
1. Locate $P$ on the diagram.
2. Observe which cell contains $P$.
3. State: dispatch from the hospital $\mathbf{\text{whose cell contains } P}$ (guaranteed nearest by construction).
Sites $A(2, 2)$ and $B(6, 10)$. Find the boundary separating them in $y=mx+c$ form.
1. Midpoint: $M = (4, 6)$.
2. Gradient $AB$: $2$; perpendicular $= -0.5$.
3. Line: $y - 6 = -0.5(x - 4)$.
4. State: $\mathbf{y = -0.5x + 8}$.
Where should a toxic dump go to be as far as possible from three towns?
1. Feature: the Voronoi vertex shared by all three cells.
2. Reasoning: it is equidistant from all three towns and is the centre of the largest empty circle.
3. State: place the dump exactly on the $\mathbf{\text{Voronoi vertex}}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
A city has two major cell phone towers, \(T_1\) at \((-2, 4)\) and \(T_2\) at \((6, -2)\). The Voronoi boundary separating their coverage areas is the perpendicular bisector of the line segment joining them.
Find the equation of this boundary line, giving your answer in the form \(ax + by + d = 0\), where \(a, b, d \in \mathbb{Z}\).
Two locations on a map are represented by the coordinates \(A(-2, 5)\) and \(B(4, 1)\).
Two sites on a coordinate map are located at \(P(2,2)\) and \(Q(8,4)\). The boundary between their Voronoi regions is the perpendicular bisector of the line segment connecting them. Find the exact equation of this boundary line in the form \(ax + by + d = 0\), where \(a, b, d \in \mathbb{Z}\).
There are 26 exam-style voronoi diagrams questions in the AI SL question bank (Paper 1: 16 · Paper 2: 10), graded 6 very hard, 9 hard, 6 medium, 4 starter, 1 easy. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 16 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.