Voronoi: notes and practice questions

Three sensor stations, Alpha (A), Beta (B), and Gamma (C), are positioned in a national park. Their coordinates are A(2, 8), B(10, 4), and C(6, 0) respectively.

The diagram below shows these points and the perpendicular bisectors of the segments connecting them.

The perpendicular bisector of the line segment [BC] intercepts the axes at coordinates (0, 10) and (10, 0).

Write down the equation of the perpendicular bisector of [BC].

The equation of the perpendicular bisector of [AB] is $y = 2x - 6$.

Find the coordinates of point V, where the three perpendicular bisectors meet. Give your answer to four significant figures.

A Voronoi diagram is constructed with sensor stations A, B, and C as the three sites.

Draw, clearly, the edges of the Voronoi diagram on the given diagram.

Three emergency service hubs are located at coordinates F1 (2, 5), P1 (10, 5), and H1 (10, 15), where distances are measured in kilometers. A central dispatch unit needs to be built at a location equidistant from these three hubs. This location is a vertex of the Voronoi diagram formed by the hubs.

Write down the coordinates of this central dispatch unit.

Find the equation of the perpendicular bisector of the segment connecting F1 (2, 5) and H1 (10, 15).

A new emergency hub, S1 (4, 8), is established, and the Voronoi diagram is redrawn. Several potential locations for a new drone charging station are at the vertices of this updated diagram. The coordinates of four such vertices are given as V_alpha (6, 10), V_beta (6, 4.5), V_gamma (8.75, 10), and V_delta ($\frac{223}{28}$, $\frac{59}{7}$). The drone charging station should be built at the vertex that is as far as possible from its nearest emergency service hub.

By calculating appropriate distances, find the location of the drone charging station.

Hence, write down the distance of the drone charging station to the nearest emergency service hub.

A team of ecologists has set up several automated sound recorders (stations) in a wildlife sanctuary. The sanctuary is mapped on a coordinate grid where each unit represents 1 km. The locations of three stations are:

Station P: $(2, 2)$

Station Q: $(10, 2)$

Station R: $(6, 8)$

Find the coordinates of the Voronoi vertex that is equidistant from stations P, Q, and R.

Consider a different section of the sanctuary with new monitoring stations. The Voronoi cell for Station E is a quadrilateral with vertices at $(4,1)$, $(7,4)$, $(4,7)$, and $(1,4)$.

(i) Calculate the area, in square units, of the Voronoi cell for Station E.

(ii) Given that one square unit on the grid represents $0.5$ km$^2$ in the sanctuary, find the area, in km$^2$, of the Voronoi cell for Station E.

In another section of the sanctuary, Station X has a Voronoi cell with an area of $20$ km$^2$, and Station Y has a Voronoi cell with an area of $10$ km$^2$. However, the ecologists recorded the same number of a specific rare bird species in both cells during the monitoring period.

Suggest a reason why the number of birds recorded is not proportional to the area of the Voronoi cell.

A telecommunications company is planning to install new signal boosters in a remote region. The optimal locations for these boosters are determined by Voronoi diagrams to ensure maximum coverage. Consider three proposed booster sites: $A(4, 8)$, $B(9, 3)$, and $C(-1, 3)$.

(a) (i) Vertex $X$ is a critical point equidistant from sites $A$, $B$, and $C$. Write down the coordinates of $X$.

(a) (ii) The exact distance from vertex $X$ to site $A$ is $\sqrt{n}$ km. Write down the value of $n$.

Consider three other proposed booster sites: $P(1, 2)$, $Q(7, 4)$, and $R(3, 8)$.

(b) (i) Vertex $Y(a, b)$ is equidistant from sites $P$, $Q$, and $R$. Write down the value of $a$.

(b) (ii) Find the exact value of $b$.

A telecommunications company is planning to install new cell towers. The locations of three proposed towers are $T_1(1, 3)$, $T_2(7, 3)$, and $T_3(4, 0)$. A fourth tower, $T_4(5, 8)$, is also planned.

(a) Find the equation of the boundary between towers $T_1$ and $T_4$. Give your answer in the form $Ax + By + C = 0$, where $A, B, C$ are integers.

A new emergency repeater station, $V(a, b)$, needs to be installed such that it is equidistant from towers $T_1(1, 3)$, $T_2(7, 3)$, and $T_3(4, 0)$.

(i) Find the value of $a$.

(ii) Find the value of $b$.

(c) The exact distance from the repeater station $V$ to tower $T_1$ is $\sqrt{n}$. Write down the value of $n$.

(a) Two proposed locations for new fire stations are $P(2, 5)$ and $Q(8, 11)$. The city wants to build a new community centre that is equidistant from both proposed fire stations. Find the equation of the perpendicular bisector of the line segment connecting $P$ and $Q$.

(b) Four existing emergency stations are located at $A(2, 5)$, $B(8, 11)$, $C(5, 2)$, and $D(10, 7)$. A new critical facility is being built at $F(6, 8)$. Determine which existing emergency station is the closest to the new facility at $F(6, 8)$. Justify your answer with calculations.

(c) An incident occurs at location $I(7, 6)$. Emergency services define a "rapid response" as being within $3.5$ km of the closest emergency station. State whether a rapid response is guaranteed for the incident at $I(7, 6)$. Justify your answer with calculations.

(a) Points $A(2, 2)$, $B(2, 8)$, $C(10, 10)$, $D(8, 4)$ and $E(6, 6)$ represent emergency service hubs in the city of Eldoria. These hubs are illustrated in the following coordinate axes.

Horizontal scale: 1 unit represents 1 km.

Vertical scale: 1 unit represents 1 km.

Calculate the gradient of the line segment $AE$.

(b) The city planner draws several straight lines to form an incomplete Voronoi diagram for the emergency service hubs.

Find the equation of the line which would complete the Voronoi cell containing hub $E$.

Give your answer in the form $ax + by + d = 0$ where $a, b, d \in \mathbb{Z}$.

(c) In the context of the question, explain the significance of the Voronoi cell containing hub $E$.

Emergency supply depots P, Q, R, S, T are located in a disaster-stricken region, with coordinates $P(2, 3)$, $Q(6, 1)$, $R(10, 5)$, $S(8, 9)$, and $T(4, 7)$. The coordinates are in kilometres.

(a) Calculate the gradient of the line segment PT.

The disaster relief team is creating a Voronoi diagram to delineate service areas for each depot.

(b) Find the equation of the line which forms part of the boundary between the service areas of depot T and depot S. Give your answer in the form $ax + by + d = 0$ where $a, b, d \in \mathbb{Z}$.

(c) In the context of this question, explain the significance of the Voronoi cell containing depot T.

The following grid shows a city's emergency service layout. There are four service stations centred at points A, B, C, and D. A Voronoi diagram for these four points could be constructed. One unit represents 1 kilometre.

Fire station A is located at $(3, 5)$.

The equation of the perpendicular bisector of the line segment $[AB]$ is $x = 7$.

(a) Write down the coordinates of fire station B.

Police station C is located at $(1, 10)$, and police station D is located at $(7, 2)$.

(b) Find

(i) the coordinates of the midpoint of the line segment $[CD]$.

(ii) the equation of the perpendicular bisector of $[CD]$. Give your answer in the form $y = mx + c$.

A city is served by four emergency service stations: Fire Station A $(3, 5)$, Fire Station B $(11, 5)$, Police Station C $(1, 10)$, and Police Station D $(7, 2)$. The average response times (in minutes) for each station are given in the table below.

Using nearest-neighbour interpolation, find the average response time for an incident occurring at point $(9, 3)$.

The city council wants to determine if the average emergency response time for fire incidents ($\mu_F$) is different from the average response time for police incidents ($\mu_P$). They decide to use a two-sample $t$-test at a $5\%$ level of significance.

For this test, the null hypothesis is:

$H_0: \mu_F = \mu_P$

(d) State the alternative hypothesis.

To gather data on response times, the city council decides to select every 10th incident report from the past year's records.

(e) (i) State which one of the following methods best describes this sampling technique:

convenience, systematic, or stratified.

(ii) State one disadvantage of this sampling technique.

The city council instead collects the data using simple random sampling.

The $p$-value for the sampled data is $0.035$.

(f) State the conclusion for the test. Justify your answer.

The telecommunications company "ConnectAll" is optimizing its cell tower network. The locations of six existing cell towers (labelled A, B, C, D, E, and F) are given by their coordinates in a local grid system, where distances are measured in kilometres.

Tower A: $(1, 9)$

Tower B: $(4, 7)$

Tower C: $(9, 8)$

Tower D: $(9, 3)$

Tower E: $(2, 2)$

Tower F: $(7, 1)$

A customer at coordinates $(3, 8)$ wants to connect to the closest cell tower. Write down the tower they should connect to.

A customer at coordinates $(0, 1)$ wants to connect to the closest cell tower. Tower E is currently offline for maintenance. Write down the tower they should connect to.

Tower C is at $(9, 8)$ and Tower D is at $(9, 3)$.

Find the equation of the perpendicular bisector of the line segment CD.

Tower B is at $(4, 7)$ and Tower C is at $(9, 8)$.

Find the equation of the perpendicular bisector of the line segment BC.

Hence find

(i) the coordinates of the point which is of equal distance from towers B, C, and D.

Hence find

(ii) the distance of this point from Tower D.