Coordinate Geometry & Voronoi (AI SL)
This topic starts with the coordinate-geometry basics of midpoints, gradients and perpendicular lines, then applies them to a genuinely visual piece of AI-only content: Voronoi diagrams. A Voronoi diagram divides a plane into regions ("cells") according to which of several fixed points ("sites") is nearest - the boundaries between cells are exactly the perpendicular bisectors you've already learned to find.
What the syllabus says
This topic maps onto two points in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL3.5 | Equations of perpendicular bisectors, given either two points, or the equation of a line segment and its midpoint. |
| SL3.6 | Voronoi diagrams: sites, vertices, edges, cells. Addition of a site to an existing diagram. Nearest-neighbour interpolation. Applications of the "toxic waste dump" problem. In examinations, the coordinates of sites for calculating perpendicular bisector equations will be given, and students will not be required to construct perpendicular bisectors by hand. The solution point (Voronoi vertex) will always be at the intersection of three edges. |
Gradient and equation-of-a-line skills used throughout are shared prior learning with SL2.1.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the midpoint formula?
The midpoint of the segment joining \((x_1,y_1)\) and \((x_2,y_2)\) is the average of the two x-coordinates and the average of the two y-coordinates: \(M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\).
e.g. For \(P(2,1)\) and \(Q(8,5)\): \(M=\left(\dfrac{2+8}{2},\dfrac{1+5}{2}\right)=(5,3)\).
What is the gradient of a line?
The gradient measures steepness: how far a line rises for every unit it runs across, \(m=\dfrac{y_2-y_1}{x_2-x_1}\). It's the key ingredient for finding a line's equation and for testing whether two lines are perpendicular.
e.g. For \(A(-1,4)\) and \(B(3,12)\): \(m=\dfrac{12-4}{3-(-1)}=\dfrac{8}{4}=2\).
What is a perpendicular bisector?
A perpendicular bisector is the line that crosses a segment at its midpoint at a right angle. Every point on it is equally distant from the segment's two endpoints - which is exactly why it forms the boundary of a Voronoi cell.
e.g. If a segment has gradient \(\tfrac23\), its perpendicular bisector has gradient \(-\tfrac32\).
What is a Voronoi diagram?
A Voronoi diagram splits the plane into cells around a set of fixed points (sites), where every point inside a cell is closer to that cell's site than to any other. The cell boundaries (edges) are straight lines - segments of perpendicular bisectors.
e.g. For sites \(A(0,0)\) and \(B(6,0)\), the boundary between their cells is the vertical line \(x=3\), since that's where distances to A and B are equal.
What is a Voronoi vertex?
A Voronoi vertex is a point equally distant from three (or more) sites - it's where three cell edges meet. You find one by solving two of the three perpendicular bisector equations simultaneously.
e.g. For three sites forming a triangle, the Voronoi vertex nearest that triangle sits at the intersection of any two of its perpendicular bisectors.
Key formulas
A handful of coordinate-geometry formulas do all the calculating - the syllabus content itself (Voronoi vocabulary and interpretation) is more about reasoning than formulas. The two tables below summarise the toolkit, and the explanations underneath go into more depth.
Formula reference
None of these are printed in the official formula booklet - they're treated as prior knowledge carried over from coordinate geometry (SL2.1), so it's worth having them ready from memory.
| Formula | Used for | Booklet? |
|---|---|---|
| \(M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\) | Midpoint of a segment | Not in the formula booklet - prior knowledge |
| \(m=\dfrac{y_2-y_1}{x_2-x_1}\) | Gradient between two points | Not in the formula booklet - prior knowledge |
| \(y-y_1=m(x-x_1)\) | Equation of a line through a point | Not in the formula booklet - prior knowledge |
| \(m_1\times m_2=-1\) | Test for perpendicular lines | Not in the formula booklet - prior knowledge |
Two ways to define a perpendicular bisector
The syllabus explicitly allows the question to give you either starting point.
| Feature | Given two points | Given midpoint and gradient of the segment |
|---|---|---|
| First step | Find the midpoint of the two points | Midpoint is already given |
| Second step | Find the gradient of the segment | Gradient of the segment is already given |
| Shared final step | Take the negative reciprocal of the gradient, then use the midpoint | Take the negative reciprocal of the gradient, then use the midpoint |
| Example | \(P(2,1)\), \(Q(8,5)\) → \(y=-1.5x+10.5\) | Midpoint \((5,3)\), segment gradient \(\tfrac23\) → \(y=-1.5x+10.5\) |
Coordinate geometry essentials
These three results feed directly into every perpendicular bisector question.
Midpoint
\[M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\]
Average the x-coordinates and average the y-coordinates separately.
Not in the formula booklet - prior knowledgeGradient
\[m=\dfrac{y_2-y_1}{x_2-x_1}\]
The change in y divided by the change in x between any two points on the line.
Not in the formula booklet - prior knowledgePerpendicular gradient
\[m_{\perp}=-\dfrac{1}{m}\]
Flip the gradient and change its sign - this is the gradient of every Voronoi edge.
Not in the formula booklet - prior knowledgeVoronoi diagram vocabulary
The Voronoi content itself is mostly about correctly interpreting a diagram you're given, using the coordinate geometry above.
Sites, cells, edges, vertices
A site is a fixed point; its cell is the region closer to it than to any other site; edges are the cell boundaries (perpendicular bisector segments); vertices are where edges meet.
Nearest-neighbour interpolation
Every point inside a cell is estimated to share the value of that cell's site - e.g. rainfall recorded at a weather station is assumed to apply to its whole cell.
The "toxic waste dump" problem
A classic application: find the point as far as possible from every site. That point is always a Voronoi vertex, since moving off it brings you closer to at least one site.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
The points are \(A(-1,\,4)\) and \(B(3,\,12)\).
(a) Find the gradient of \([AB]\).
(b) Find the equation of the line through \(A\) and \(B\).
Worked solution
(a) Gradient \(=\dfrac{12-4}{3-(-1)}=\dfrac84\) M1
\(=2.\) A1
(b) \(y-4=2(x+1)\Rightarrow y\) M1
\(=2x+6.\) A1
Two towns are at \(P(2, 1)\) and \(Q(8, 5).\)
(a) Find the midpoint.
(b) Find the gradient of \(PQ.\)
(c) Find the equation of the perpendicular bisector.
Worked solution
(a) \(M = \left(\tfrac{2+8}{2}, \tfrac{1+5}{2}\right) = (5, 3).\) M1 A1
Step 2 - Gradient of \(PQ\).
(b) \(m_{PQ} = \dfrac{5-1}{8-2} = \dfrac{4}{6} = \tfrac23.\) M1 A1
Perpendicular gradient \(= -\tfrac32\), through \(M(5,3)\): M1
\(y - 3 = -\tfrac32(x - 5) \Rightarrow y = -\tfrac32 x + \tfrac{15}{2} + 3 = -\tfrac32 x + \tfrac{21}{2}.\) A1
(c) \(y = -\tfrac32 x + \tfrac{21}{2}.\)
The bisector is the boundary between the two towns' catchment regions.
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting to take the negative reciprocal. The perpendicular bisector does not share the segment's gradient - flip it and change its sign, or the "bisector" you find won't actually be perpendicular.
- Averaging the midpoint coordinates incorrectly. The midpoint formula adds the two x-values (and the two y-values) and halves the result - subtracting them, or halving only one coordinate, gives a point nowhere near the segment.
- Assuming a Voronoi cell boundary is curved. Every edge in a Voronoi diagram is a straight line - a segment of a perpendicular bisector - never an arc.
- Finding only two bisectors and stopping. A Voronoi vertex is equidistant from three sites, so a full solution should confirm the point lies on all three relevant edges, not just the two you happened to intersect first.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Once you have a perpendicular bisector's equation, checking it at a specific x-value is a quick way to verify a point lies on the boundary, or to read off a y-value the question asks for.
- Enter the bisector's equation as a function, then evaluate it at any x-value.
- Method 1 - Table: 2nd → GRAPH (TABLE), scroll to the x you want. Method 2 - Trace: press TRACE, then type the x-value and press ENTER. Method 3 - Home screen: type Y1(value) e.g. Y1(3) using VARS → Y-VARS → 1:Function.TI-84
- On the graph, press TRACE and type the x-value, then ENTER. Or on a Calculator page type f1(3) to evaluate the stored function at x = 3.Nspire
- Press TRACE (F1) on the graph, then type the x-value and EXE. Or use the Table view (MENU → Table) to see multiple values.Casio
Tip: Trace gives an approximate value by cursor position - typing the x-value after pressing TRACE gives the exact value.
Tabulating a perpendicular bisector's equation gives you several clean coordinate pairs to plot when sketching a Voronoi edge by hand.
- Enter the function in the Y= / graph editor.
- 2nd → WINDOW (TBLSET) to set the start and step, then 2nd → GRAPH (TABLE) to view it.TI-84
- Add a Graphs or Lists & Spreadsheet page, then menu → Table (or ctrl+T).Nspire
- Enter the function in Table mode, set the range, then TABL (F6).Casio
Tip: Use a small step (e.g. 1) to get several tidy points to plot along the bisector.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Coordinate geometry and Voronoi questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between a perpendicular bisector and a Voronoi edge?
They're the same line - a Voronoi edge is just the part of a perpendicular bisector that actually forms a boundary between two neighbouring cells. The full bisector extends infinitely; the Voronoi edge is the relevant segment of it.
How do I find a Voronoi vertex?
A Voronoi vertex is equidistant from three sites, so it's where three perpendicular bisectors meet. In practice you find it by solving any two of the three bisector equations simultaneously - the third will pass through the same point.
Do I need to construct perpendicular bisectors from scratch?
No. The IB syllabus explicitly says students will not be required to construct perpendicular bisectors by hand in a Voronoi context - exam questions give you the site coordinates and ask you to find the bisector equation, or give you the equations directly.
Is the distance formula given in the formula booklet?
The midpoint and gradient formulas used on this page are treated as prior knowledge and are not printed in the formula booklet, so it's worth being able to derive and apply them quickly from memory.
Sub-topics
Coordinate Geometry & Voronoi broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI SL syllabus unit, in case you want to keep going.