Voronoi Construction (AI HL)
Before you can use a Voronoi diagram, you need to build the pieces it's made of: the perpendicular bisector between two sites, and the vertex where three bisectors meet. Exam questions give you site coordinates and ask you to derive these algebraically - you're never asked to physically draw the diagram by hand. This page focuses on that construction step. It's part of the broader Voronoi Diagrams topic.
18 questions on this sub-topic.
Key formula
Covered under IB syllabus reference SL3.6: sites, vertices, edges and cells, with site coordinates always given so the bisector equations can be found algebraically.
Perpendicular bisector between two sites
Find the midpoint of the two sites and the negative reciprocal of the gradient between them, then write the line through that midpoint with that gradient. This line is the shared edge of the two sites' cells.
Not in the formula booklet - coordinate geometryFinding a Voronoi vertex
Solve two bisector equations simultaneously - the intersection is equidistant from all three sites involved, and is always where exactly three edges meet in exam questions.
Not in the formula booklet - simultaneous equationsNeed the full syllabus wording and formula reference table? See Voronoi Diagrams.
Worked examples
Find the perpendicular bisector of \(A(2, 3)\) and \(B(2, 9).\)
Worked solution
The segment is vertical, M1
so the bisector is horizontal A1
through the midpoint \((2, 6)\): M1
\(y = 6.\) A1
Find the point equidistant from the three sites \(A(0,0)\), \(B(6,0)\) and \(C(0,4)\) (the Voronoi vertex).
Worked solution
Equidistant from \(A, B\): bisector \(x\) M1
\(= 3.\) A1
Equidistant from \(A, C\): bisector \(y\) M1
\(= 2,\) so the vertex is \((3, 2).\) A1
Find the perpendicular bisector of \(A(-2, 4)\) and \(B(4, -2).\)
Worked solution
Midpoint \((1, 1).\) M1
Midpoint \((1, 1).\) A1
Gradient \(AB = -1\), bisector gradient \(= 1.\) M1
Bisector gradient \(= 1.\) A1
\(y = x.\) A1
Common mistakes
- Using the gradient between the sites instead of its negative reciprocal. A bisector is perpendicular to the segment joining the two sites, so the gradient must be flipped and inverted - reusing the original gradient is the single most common slip in these questions.
- Averaging only one coordinate when finding the midpoint. The midpoint needs both \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) coordinates averaged separately - dropping one and copying the other across gives a point that isn't on the bisector at all.
- Stopping after finding one bisector when a vertex is asked for. A Voronoi vertex needs two bisector equations solved simultaneously - a single bisector only gives an edge, not the point where three cells meet.
Ready to practise properly?
18 Voronoi-construction questions, marked instantly like the real exam.
Quick answers
How do I find the perpendicular bisector between two sites?
Find the midpoint of the two sites and the negative reciprocal of the gradient between them, then write the equation of the line through that midpoint with that gradient.
How do I find a Voronoi vertex from bisector equations?
Solve two bisector equations simultaneously. Their intersection is equidistant from all three sites involved, and in exam questions it is always the point where exactly three edges meet.