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Date	May 2021	Marks available	2	Reference code	21M.1.SL.TZ1.5
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Write down	Question number	5	Adapted from	N/A

Question

The Voronoi diagram below shows three identical cellular phone towers, $T 1, T 2$ $T 1, T 2$ and $T 3$ $T 3$ . A fourth identical cellular phone tower, $T 4$ $T 4$ is located in the shaded region. The dashed lines in the diagram below represent the edges in the Voronoi diagram.

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

Tim stands inside the shaded region.

Tower $T2$ $T2$ has coordinates $(- 9, 5)$ $(- 9, 5)$ and the edge connecting vertices $A$ $A$ and $B$ $B$ has equation $y = 3$ $y = 3$ .

Explain why Tim will receive the strongest signal from tower $T4$ $T4$ .

[1]

Write down the coordinates of tower $T4$ $T4$ .

[2]

Tower $T1$ $T1$ has coordinates $(- 13, 3)$ $(- 13, 3)$ .

Find the gradient of the edge of the Voronoi diagram between towers $T1$ $T1$ and $T2$ $T2$ .

[3]

Markscheme

every point in the shaded region is closer to tower $T4$ $T4$ R1

Note: Specific reference must be made to the closeness of tower $T4$ $T4$ .

[1 mark]

$(- 9, 1)$ $(- 9, 1)$ A1A1

Note: Award A1 for each correct coordinate. Award at most A0A1 if parentheses are missing.

[2 marks]

correct use of gradient formula (M1)

e.g. $(m =) \frac{5 - 3}{- 9 - - 13} (= \frac{1}{2})$ $(m =) \frac{5 - 3}{- 9 - - 13} (= \frac{1}{2})$

taking negative reciprocal of their $m$ $m$ (at any point) (M1)

edge gradient $= - 2$ $= - 2$ A1

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.6—Voronoi diagrams

Show 27 related questions

22M.3.AHL.TZ1.2c.i:
Find the equation of the perpendicular bisector of $[IF]$ $[IF]$ .
22M.3.AHL.TZ1.2b:
Sketch a Voronoi diagram showing the regions within the metropolitan area that are closest to each town.
22M.3.AHL.TZ1.2c.iii:
Sketch this new Voronoi diagram showing the regions within the metropolitan area which are closest to each town.
EXM.2.SL.TZ0.4a:
Sketch a Voronoi diagram to represent this information.
22M.3.AHL.TZ1.2e.ii:
Make one possible criticism of the council’s choice of location.
22M.3.AHL.TZ1.2c.ii:
Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$ $(20, 37.5)$ , find the coordinates of the other two vertices within the metropolitan area.
22M.3.AHL.TZ1.2e.i:
Find the location of the toxic waste dump, given that this location is not on the edge of the metropolitan area.
21M.1.SL.TZ1.5a:
Explain why Tim will receive the strongest signal from tower $T4$ $T4$ .
21N.1.SL.TZ0.7b.i:
Write down the equation of the perpendicular bisector of $[PC]$ $[PC]$ .
EXN.2.SL.TZ0.5e:
A new office is to be built within the triangle formed by $A$ $A$ , $B$ $B$ and $C$ $C$ , at a point as far as possible from the other three offices.

Find the distance of this office from each of the other offices.
22M.2.SL.TZ2.3a:
Find the midpoint of $[BD]$ $[BD]$ .
22M.3.AHL.TZ1.2a:
The model assumes that each town is positioned at a single point. Describe possible circumstances in which this modelling assumption is reasonable.
EXM.2.SL.TZ0.4b:
Sketch a new Voronoi diagram to represent this new situation.
EXM.2.SL.TZ0.4c:
State what the shape of the land, owned by the youngest child, is.
SPM.1.SL.TZ0.7c:
In the context of the question, explain the significance of the Voronoi cell containing site E.
SPM.1.SL.TZ0.7a:
Calculate the gradient of the line segment AE.
EXM.2.SL.TZ0.4e:
Find how much land an older child has lost.
EXM.1.SL.TZ0.4:
The diagram below is part of a Voronoi diagram.

Diagram not to scale

A and B are sites with B having the co-ordinates of (4, 6). L is an edge; the equation of this perpendicular bisector of the line segment from A to B is $y = - 2 x + 9$ $y = - 2x + 9$

Find the co-ordinates of the point A.
EXM.2.SL.TZ0.4f:
State, with a reason, if all five children now own an equal amount of land.
EXN.2.SL.TZ0.5d:
State a further assumption that must be made in order to use area covered as a measure of whether or not the manager of office $C$ $C$ is responsible for more students than the managers of offices $A$ $A$ and $B$ $B$ .
21M.1.SL.TZ1.5c:
Tower $T1$ $T1$ has coordinates $(- 13, 3)$ $(- 13, 3)$ .

Find the gradient of the edge of the Voronoi diagram between towers $T1$ $T1$ and $T2$ $T2$ .
SPM.1.SL.TZ0.7b:
Find the equation of the line which would complete the Voronoi cell containing site E.

Give your answer in the form $a x + b y + d = 0$ $ax + by + d = 0$ where $a$ $a$ , $b$ $b$ , $d \in \mathbb{Z}$ .
EXM.2.SL.TZ0.4d:
Find the area of the youngest child’s land.
EXN.2.SL.TZ0.5b:
Find the area of campus managed by office $C$ .
EXN.2.SL.TZ0.5c:
Hence or otherwise find the areas managed by offices $A$ and $B$ .
21N.1.SL.TZ0.7a:
Show that the new station should be built at $P$ .
21N.1.SL.TZ0.7b.ii:
Hence draw the missing boundaries of the region around $P$ on the following diagram.

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Topic 3—Geometry and trigonometry

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