Voronoi Diagrams (DP IB Applications & Interpretation (AI): SL): Exam Questions

The points $\mathrm{A}(4,3),\mathrm{B}(-2,-1)$ and $\mathrm{C}(2,-5)$ on the Voronoi diagram below represent the locations of three gold coins in a computer game. Points are scored for each gold coin that is collected by an avatar. Each step that is moved reduces the energy levels of the avatar so it is advised to collect the coin that is closest to you. Each horizontal and vertical unit indicated on the graph represents 1 cm.

Your avatar is located at $(-1,3)$. 

(i) State which gold coin is closest to you.

(ii) Calculate the distance from your avatar to the nearest gold coin.

A new gold coin appears at $\mathrm{D}(-2,4)$.

Redraw the Voronoi diagram on the grid above to incorporate the new gold coin.

Show that the new gold coin is now the closest to your avatar.

A town is built in a rectangular area bounded by the lines $y=-2,y=10,x=-4$ and $x=10$ as shown on the Voronoi diagram below.

Each horizontal and vertical unit on the grid below represents 1 km.

Points $\mathrm{A}(-2,0),\mathrm{B}(4,8)$ and $\mathrm{C}(8,0)$ represent the locations of first aid responders in the town.

Calculate the area for which the first aid responder at $\mathrm{C}$ has responsibility.

A new station is due to open at $\mathrm{D}(8,2)$ and as such the Voronoi diagram will need to be adjusted.

(i) Write down the equation of the perpendicular bisector of [$\mathrm{BD}$]. Give your answer in the form $y=mx+c$.

(ii) Write down the equation of the perpendicular bisector of [$\mathrm{CD}$]. Give your answer in the form $y=mx+c$.

You have been challenged to stay in a tiger infested jungle overnight. The incomplete Voronoi diagram below shows the known position of four tigers. Each unit on the diagram represents 400 m on the ground.

Find the equation of the line that would complete the Voronoi cell containing the tiger at point D. Give your answer in the form $ax+by+d=0$ where $a,b,d\in \mathrm{\mathbb{Z}}$.

The equation of the perpendicular bisector of [$\mathrm{AD}$] is $x-2y+4=0$. Determine the coordinates of:

(i) Point P.

(ii) Point Q.

State which of the two points, $\mathrm{P}$ or $\mathrm{Q}$, is the location that is between the tigers but as far as possible from them and calculate the distance that would be between you and the nearest tiger.

In a square city measuring 15 km by 15 km, there are sixteen post boxes in a grid formation that are spaced at 5 km intervals.

Draw a sketch of a Voronoi diagram showing how the area of the city is served by the different post boxes.

Calculate the size of the largest and the smallest areas that are each served by a single post box.

It is decided that the number of post boxes in the city will be reduced and re-arranged. There are now only five post boxes available. One is located at the very centre of the city with the other four located at a distance of 5 km North, South, East and West of the centre.

 Draw a sketch of the new Voronoi diagram that would describe this situation.

Calculate the largest area that is now served by a single post box.

An incomplete Voronoi diagram of a region on the moon’s surface is shown below. Each of the five points $\mathrm{A}(-7,1),\mathrm{B}(-3,5),\mathrm{C}(-2,1)$, $\mathrm{D}(5,-3)$and $\mathrm{E}(a,b)$  represent the flag of a country that has claimed a portion of the moon. One of the points is missing from the diagram.

Write down the coordinates of the missing flag, $\mathrm{E}$, and complete the diagram.

Point $\mathrm{P}$ has coordinates $\mathrm{P}\left(-\frac{9}{2},\frac{5}{2}\right)$.

Explain the significance of the location of point $\mathrm{P}$.

Countries $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{E}$ are upset that country $\mathrm{D}$ has obtained the largest area and want $\mathrm{D}$ to reposition their flag.

 Explain how the territories claimed by $\mathrm{C}$ and $\mathrm{E}$ will be affected if $\mathrm{D}$ moves its flag to $(3,-3)$.

The diagram below shows how a square area of land has been divided up for the houses of three families at points $\mathrm{A}(1,8),\mathrm{B}(7,8)$ and $\mathrm{C}(9,4)$. Perpendicular bisectors between the houses have been used to form the boundaries of the land surrounding each house. One unit on the diagram corresponds to 10 m in real life.

Calculate the area of land that each family own.

The children from house $\mathrm{A}$ and house $\mathrm{B}$ fall in love and decide to build a house together in the area between their families.

By redrawing the Voronoi diagram, show that by building their house at $\mathrm{D}(4,8)$, the children will reduce the land surrounding house $\mathrm{A}$ more than the land surrounding house $\mathrm{B}$.

The family at house $\mathrm{C}$ complain that with the adjustment of the boundary lines, they lose 3.25% of their land. Show that this statement is true.

Three phone masts are erected in a triangular arrangement with the vertices of the triangle being A$(-1,12),B(7,6)$and $C(1,2)$. A scientist is conducting some experiments in the area is using some sensitive electronic equipment and wishes to find a suitable location. The location that he chooses must be as far as possible from each of the three phone masts in order to minimise any potential interference. One unit on the graph represents 200 m.

Draw a Voronoi diagram on the grid below.

Find the coordinates of the best position for the scientist to work.

Calculate the distance that the scientist will be from the nearest phone mast.

The local government of an area wishes to install tsunami warning sirens in order to be able to alert residents in the case of natural disasters.

A Voronoi diagram is drawn to find the appropriate locations for the warning sirens to be installed, an incomplete copy can be seen below. Five towns lie at points  $\mathrm{A}(3,6),\mathrm{B}(5,5),\mathrm{C}(6,-2),$$\mathrm{D}(2,-1.5)$  and $\mathrm{E}(-1,2)$. The points of intersection of the perpendicular bisectors are P, Q and R. One unit on the grid represents 1 km.

Write down the equation of the line that will complete the cell containing town $\mathrm{B}$. Give your answer in the form $ax+by+d=0$ where $a,b,d\in \mathrm{\mathbb{Z}}$.

Each warning siren can be heard up to a distance of 4 km.

State the minimum number of sirens required to reach all of the towns and write down their coordinates.

A geography student wishes to observe the microclimate of their back garden to see if there are differences in how much rain falls in different areas. She places a rain gauge at four separate sites. The student decides to construct a Voronoi diagram, assuming that the rain collected at one site will be the same throughout the cell in which it is situated. The positions of the sites are indicated on the diagram below at point $\mathrm{A}(2,3),\mathrm{B}(7,5),\mathrm{C}(1,8)$ and $\mathrm{D}(9,10)$. One unit on the grid represents 10 m.

Find the coordinates of the points $\mathrm{P}$ and $\mathrm{Q}$ where the cell boundaries intersect.

The rainfall at the different sites over a 24-hour period is recorded in the table below.

Give a reason that could explain why the rainfall at site C is different to the rainfall measured at the other sites.

The Voronoi diagram below shows the location of four whirlpools at points $\mathrm{A}(\mathrm{x},\mathrm{y}),\mathrm{B}(1,11),\mathrm{C}(8,4)$ and $\mathrm{D}(-3,-1)$. One unit on the diagram represents 500 m.

The perpendicular bisector of [$\mathrm{AB}$] has the equation $4x-y+24=0$.

Find the coordinates,$(x,y),$ at point $\mathrm{A}$.

A viewing station is to be constructed between the four whirlpools. The perpendicular bisector of [$\mathrm{BC}$] has the equation $y=x+3$ and the perpendicular bisector of [$\mathrm{BD}$] has the equation $x+3y-14=0$.

(i)      Find the coordinates of the point which is furthest from the whirlpools and

         therefore the safest location in which to construct the station.

(ii)     Write down the distance from the viewing station to the nearest whirlpool.

A group of four pirates claim a circular island of radius 8 km, with the centre of the island located at point $\mathrm{P}(0,0)$ on their map. One unit on the map represents 1 km.

The pirates want to set up their bases with boundaries that are formed by finding the perpendicular bisectors between the pirates.

Write down the coordinates of the locations of each of the pirates’ bases if they are all to have the same area of land in their cell.

Calculate the area of land that is in each pirate’s cell.

A fifth pirate, the Captain who had been lost at sea, arrives at the island and establishes his base at point $\mathrm{P}$. None of the other pirates move their bases.

Calculate the percentage of their land that each of the original four pirates has to hand over to the Captain.

An energy company supplies five towns with gas in an area of 20 km by 20 km. The Voronoi diagram below shows the location of the towns at $\mathrm{A}(-8,8),\mathrm{B}(2,6),\mathrm{C}(0,-2),\mathrm{D}(-6,-8)$ and $\mathrm{E}(-14,-2)$. The boundaries of the cells are the main pipelines running between the towns, to which the towns will be connected.

The perpendicular bisector of [$\mathrm{AE}$] has the equation $3x+5y+18=0$

The perpendicular bisector of [$\mathrm{AC}$] has the equation $4x-5y+31=0$

The perpendicular bisector of [$\mathrm{DE}$] has the equation $4x-3y+25=0$

Each unit on the map represents 1 km.

Calculate the total length of the main pipelines in the area.