Voronoi Diagrams (DP IB Maths: AI HL)

Exam Questions

3 hours18 questions
1a
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4 marks

Points straight A left parenthesis negative 2 comma negative 7 right parenthesis comma straight B left parenthesis 1 comma 4 right parenthesis comma straight C left parenthesis negative 4 comma negative 1 right parenthesis and straight D left parenthesis 5 comma 0 right parenthesis on the Voronoi diagram below represent the locations of four cinemas in Berlin, Germany.

Horizontal scale: 1 unit represents 1 km.  Vertical scale: 1 unit represents 1 km.

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Amy wants to go to a cinema and her house is located at (-1,1).

(i)
Determine which cinemas Amy’s house is closest to.

(ii)
Calculate the distance from Amy’s house to these two cinemas.
1b
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2 marks

Kayla’s apartment is an equal distance from cinemas straight A and straight C.

Find the coordinates at which Kayla's apartment is located.

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2a
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1 mark

Sites straight A, straight B and straight C on the Voronoi diagram below represent the location of solar panels. Horizontal scale: 1 unit represents 10 space km.  Vertical scale: 1 unit represents 10 space km.

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A fourth site, straight D, is missing from the diagram. Write down the coordinates for site straight D.

2b
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4 marks

The perpendicular bisectors surrounding straight B intersect at points left parenthesis 0 comma 4 right parenthesis comma left parenthesis 6.375 comma 4 right parenthesis spaceand left parenthesis 5.1 comma 9.1 right parenthesis. Calculate the area of cell straight B.

2c
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2 marks

A point straight E is located at (5, 10). Find the distance from straight E to the nearest solar panel.

2d
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1 mark

The daily average number of watts produced by each solar panel is given in the table below.

Site A B C D
Watts per day 276 293 312 322

Estimate the watts produced at straight F left parenthesis 9 comma 1 right parenthesis.

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3a
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2 marks

Points straight A left parenthesis 0.5 comma 1.5 right parenthesis comma straight B left parenthesis 1.5 comma 5.5 right parenthesis comma straight C left parenthesis 2.5 comma 3.5 right parenthesis comma straight D left parenthesis 4.5 comma 4.5 right parenthesis and straight E left parenthesis 5.5 comma 0.5 right parenthesis represent mechanics in a city. The mechanics are shown below on an incomplete Voronoi diagram.

Horizontal scale:1  unit represents 1 space km. Vertical scale: 1 unit represents 1 space km.

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Calculate the gradient of the line segment CD.

3b
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3 marks

Find the equation of the line which would complete the Voronoi cell containing site space straight C. Give your answer in the form a x plus b y plus d equals 0 where a comma b comma d∈ℤ.

3c
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1 mark

In the context of the question, explain the significance of the Voronoi cell containing
site straight C.

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4a
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2 marks

Rangers use aerial imagery to help locate wolfpacks in Yellowstone National Park. This week the plane is not available so they must use last week’s image which shows the last known locations of five wolfpacks at points straight A left parenthesis 1 comma 4 right parenthesis comma straight B left parenthesis 3 comma 9 right parenthesis comma straight C left parenthesis 5 comma 5 right parenthesis comma straight D left parenthesis 8 comma 2 right parenthesis andstraight E left parenthesis 9 comma 10 right parenthesis space as illustrated on the following coordinate axes.

Horizontal scale: 1 unit represents 10 space km. Vertical scale: 1 unit represents 10 space km.

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Wolfpacks stick to very rigid territories keeping their distance from other packs to avoid confrontation. Using the ariel image, rangers draw three straight lines to form an incomplete Voronoi diagram.

Calculate the gradient of the line segment BC.

4b
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3 marks

Find the equation of the line which would complete the Voronoi cell containing site space straight C. Give your answer in the form a x plus b y plus d equals 0 where a comma b comma d∈ ℤ.

4c
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3 marks

There is one straight line missing from the Voronoi diagram below.

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Find the equation of the missing line. Give your answer in the form a x plus b y plus d equals 0 where a comma b comma d space∈ ℤ

4d
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1 mark

A family of rabbits are sighted at point straight F(10, 5).

Write down the wolfpack territory the family of rabbits are in.

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5a
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5 marks

Two schools are represented by points straight A left parenthesis 3 comma 15 right parenthesis and straight B left parenthesis 6 comma 27 right parenthesis on the graph below. A road, represented by the line space straight R with equation y minus 2 x minus 2 equals 0, passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.

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Find the equation of the perpendicular bisector of [AB]. Give your equation in the

form a x plus b y plus d equals 0 where a comma b comma d ∈ ℤ.

5b
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2 marks

Determine the coordinates of the point on straight R where the bus stop should be located.

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6a
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2 marks

The sites straight A comma straight B, straight C and straight D in the Voronoi diagram below represent the locations of active volcanos in an Indonesian region and points straight E and straight F are intersections.

Horizontal scale: 1 unit represent 10 space km. Vertical scale: 1 unit represent 10 space km.

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There is a population centre at Pleft parenthesis 7 comma 5 right parenthesis .

Calculate the distance from the population centre to the closest volcano.

6b
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3 marks

Determine the optimal position for a central shopping centre in the region, such that it is as far away from a volcano as possible. 

6c
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3 marks

A geologist says a safe distance for a shopping centre from a volcano is 45 space km. Determine whether the position of the shopping centre will be safe.

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1a
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3 marks

The points straight A left parenthesis 4 comma 3 right parenthesis comma straight B left parenthesis negative 2 comma negative 1 right parenthesis and straight C left parenthesis 2 comma negative 5 right parenthesis 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.

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Your avatar is located at left parenthesis negative 1 comma 3 right parenthesis. 

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

(ii)
Calculate the distance from your avatar to the nearest gold coin.
1b
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4 marks

A new gold coin appears at straight D left parenthesis negative 2 comma 4 right parenthesis.

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

1c
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2 marks

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

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2a
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3 marks

A town is built in a rectangular area bounded by the lines y equals negative 2 comma y equals 10 comma x equals negative 4 and space x equals 10 as shown on the Voronoi diagram below.

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

Points straight A left parenthesis negative 2 comma 0 right parenthesis comma straight B left parenthesis 4 comma 8 right parenthesis and straight C left parenthesis 8 comma 0 right parenthesis represent the locations of first aid responders in the town.

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Calculate the area for which the first aid responder at straight C has responsibility.

2b
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3 marks

A new station is due to open at straight D left parenthesis 8 comma 2 right parenthesis and as such the Voronoi diagram will need to be adjusted.

(i)
Write down the equation of the perpendicular bisector of [BD]. Give your answer in the form y equals m x plus c.

(ii)
Write down the equation of the perpendicular bisector of [CD]. Give your answer in the form y equals m x plus c.

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3a
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4 marks

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.

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Find the equation of the line that would complete the Voronoi cell containing the tiger at point D. Give your answer in the form a x plus b y plus d equals 0 where a comma b comma d element of straight integer numbers.

3b
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4 marks

The equation of the perpendicular bisector of [AD] is x minus 2 y plus 4 equals 0. Determine the coordinates of:

(i)
Point P.

(ii)
Point Q.
3c
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3 marks

State which of the two points, straight P or straight 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.

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4a
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2 marks

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.

4b
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2 marks

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

4c
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2 marks

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.

4d
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2 marks

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

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5a
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2 marks

An incomplete Voronoi diagram of a region on the moon’s surface is shown below. Each of the five points straight A left parenthesis negative 7 comma 1 right parenthesis comma straight B left parenthesis negative 3 comma 5 right parenthesis comma straight C left parenthesis negative 2 comma 1 right parenthesis, straight D left parenthesis 5 comma negative 3 right parenthesis spaceand straight E left parenthesis a comma b right parenthesis  represent the flag of a country that has claimed a portion of the moon. One of the points is missing from the diagram.

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Write down the coordinates of the missing flag, straight E, and complete the diagram.

5b
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1 mark

Point straight P has coordinates straight P open parentheses negative 9 over 2 comma 5 over 2 close parentheses.

Explain the significance of the location of point straight P.

5c
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3 marks

Countries straight A comma straight B comma straight C and straight E are upset that country straight D has obtained the largest area and want straight D to reposition their flag.

 Explain how the territories claimed by straight C and straight E will be affected if straight D moves its flag to left parenthesis 3 comma negative 3 right parenthesis.

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6a
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4 marks

The diagram below shows how a square area of land has been divided up for the houses of three families at points straight A left parenthesis 1 comma 8 right parenthesis comma straight B left parenthesis 7 comma 8 right parenthesis and straight C left parenthesis 9 comma 4 right parenthesis. 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.

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Calculate the area of land that each family own.

6b
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4 marks

The children from house straight A and house straight 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 straight D left parenthesis 4 comma 8 right parenthesis, the children will reduce the land surrounding house straight A more than the land surrounding house straight B.

6c
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3 marks

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

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1a
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4 marks

Three phone masts are erected in a triangular arrangement with the vertices of the triangle being Aleft parenthesis negative 1 comma 12 right parenthesis comma B left parenthesis 7 comma 6 right parenthesis space spaceand C left parenthesis 1 comma 2 right parenthesis. 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.

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1b
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2 marks

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

1c
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2 marks

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

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2a
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4 marks

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  straight A left parenthesis 3 comma 6 right parenthesis comma straight B left parenthesis 5 comma 5 right parenthesis comma straight C left parenthesis 6 comma negative 2 right parenthesis commastraight D left parenthesis 2 comma negative 1.5 right parenthesis  and straight E left parenthesis negative 1 comma 2 right parenthesis. The points of intersection of the perpendicular bisectors are P, Q and R. One unit on the grid represents 1 km.

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Write down the equation of the line that will complete the cell containing town straight B. Give your answer in the form a x plus b y plus d equals 0 where a comma b comma d element of straight integer numbers.

2b
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6 marks

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.

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3a
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5 marks

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 straight A left parenthesis 2 comma 3 right parenthesis comma straight B left parenthesis 7 comma 5 right parenthesis comma straight C left parenthesis 1 comma 8 right parenthesis and straight D left parenthesis 9 comma 10 right parenthesis. One unit on the grid represents 10 m.

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Find the coordinates of the points straight P and straight Q where the cell boundaries intersect.

3b
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1 mark

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

  Site Site A Site B Site C Site D
  Rainfall (mm) 22 24 3 26

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

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4a
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6 marks

The Voronoi diagram below shows the location of four whirlpools at points straight A left parenthesis straight x comma straight y right parenthesis comma straight B left parenthesis 1 comma 11 right parenthesis comma straight C left parenthesis 8 comma 4 right parenthesis and straight D left parenthesis negative 3 comma negative 1 right parenthesis. One unit on the diagram represents 500 m.

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The perpendicular bisector of [AB] has the equation 4 x minus y plus 24 equals 0.

Find the coordinates,left parenthesis x comma y right parenthesis comma space at point straight A.

4b
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4 marks

A viewing station is to be constructed between the four whirlpools. The perpendicular bisector of [BC] has the equation y equals x plus 3 and the perpendicular bisector of [BD] has the equation x plus 3 y minus 14 equals 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.

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5a
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4 marks

A group of four pirates claim a circular island of radius 8 km, with the centre of the island located at point space straight P left parenthesis 0 comma 0 right parenthesis 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.

5b
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2 marks

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

5c
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4 marks

A fifth pirate, the Captain who had been lost at sea, arrives at the island and establishes his base at point straight 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.

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6
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15 marks

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 straight A left parenthesis negative 8 comma 8 right parenthesis comma straight B left parenthesis 2 comma 6 right parenthesis comma straight C left parenthesis 0 comma negative 2 right parenthesis comma straight D left parenthesis negative 6 comma negative 8 right parenthesis and straight E left parenthesis negative 14 comma negative 2 right parenthesis. The boundaries of the cells are the main pipelines running between the towns, to which the towns will be connected.

The perpendicular bisector of [AE] has the equation 3 x plus 5 y plus 18 equals 0

The perpendicular bisector of [AC] has the equation 4 x minus 5 y plus 31 equals 0

The perpendicular bisector of [DE] has the equation 4 x minus 3 y plus 25 equals 0

Each unit on the map represents 1 km.

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

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