SL 3.6—Voronoi diagrams Questionbank

Practice SL 3.6—Voronoi diagrams with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A Voronoi diagram of four fast-food restaurants is shown. A new customer appears at point $P$ .

List all restaurants that are equally closest to point $P$ .

[1]

If one restaurant is moved from $(2,6)$ to $(3,6)$ , how does this affect the Voronoi region?

[2]

Question 2

SLPaper 1

Four schools are located at the vertices of a square: $A(2,2), B(6,2), C(6,6)$ , and $D(2,6)$ . A fifth school is to be added so that its Voronoi region shares boundaries with all four existing regions.

Explain the geometric condition required for this to occur.

[2]

Give a set of coordinates for the fifth school and justify why it satisfies the condition.

[2]

Sketch the updated Voronoi diagram.

[2]

Question 3

SLPaper 1

A Voronoi diagram is constructed using points $A(1,1), B(5,1)$ , and $C(3,4)$ . A new point $D(2,2)$ is added.

Sketch the modified Voronoi diagram including point $D$ .

[2]

Describe how the region of point $A$ will change.

[1]

Question 4

SLPaper 1

A city has three hospitals located at points $H_{1}(1,2), H_{2}(5,2)$ , and $H_{3}(3,5)$ . A Voronoi diagram is created to determine which areas are closest to each hospital.

Plot the hospitals and sketch a rough Voronoi diagram.

[2]

A person at point $P(3,3)$ needs medical attention. Which hospital should they go to?

[1]

Question 5

SLPaper 1

A cell tower company is placing three towers at $T_{1}(2,4), T_{2}(6,4)$ , and $T_{3}(4,1)$ . A customer is located at point $C(4,3)$ .

Construct a Voronoi diagram for the tower locations.

[2]

Determine which tower the customer connects to.

[1]

Justify your answer using distance calculations.

[2]

Question 6

SLPaper 1

The diagram below shows a Voronoi diagram formed by five points labeled $A$ , $B, C, D$ , and $E$ . Each region corresponds to the area closest to one of the points.

Which region corresponds to point $C$ ?

[1]

What is the perpendicular bisector used to separate regions of $A$ and $B$ ?

[2]

Question 7

SLPaper 1

The Voronoi diagram below shows regions around delivery centers $X, Y$ , and $Z$ . Point $P$ lies within the region of $Y$ .

Verify that $P$ is closer to $Y$ than to $X$ or $Z$ .

[3]

Question 8

SLPaper 1

Consider points $A(2,3)$ and $B(6,5)$ . A region $\mathcal{R}$ is defined as all points $P(x, y)$ that are closer to $A$ than to $B$ and lie within the circle $(x-4)^{2}+(y-4)^{2}=9$ .

Find the equation of the perpendicular bisector of $A B$ .

[2]

Determine the inequality that defines the half-plane of $\mathcal{R}$ (closer to $A$ ).

[2]

Determine the area of region $\mathcal{R}$ using integration or geometry.

[3]

Question 9

SLPaper 1

A Voronoi diagram is created using four shops located at $S_{1}(2,2), S_{2}(6,2)$ , $S_{3}(6,6)$ , and $S_{4}(2,6)$ .

Without plotting, explain why the region for $S_{1}$ is bounded.

[1]

Suggest coordinates for a point that would fall within the region of $S_{1}$ .

[1]

Add a new shop at $S_{5}$ so that it reduces the region of $S_{1}$ significantly. State possible coordinates and justify.

[2]

Question 10

SLPaper 1

A mobile service company places towers at $P(0,0), Q(6,0)$ , and $R(3,6)$ . A user is located at point $U(x, y)$ .

Derive the inequalities that define the region where the user would be closest to tower $R$ .

[4]

Determine whether the user at $U(3,3)$ is in this region.

[2]