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Date	May Specimen paper	Marks available	2	Reference code	SPM.1.SL.TZ0.7
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Calculate	Question number	7	Adapted from	N/A

Question

Points A(3, 1), B(3, 5), C(11, 7), D(9, 1) and E(7, 3) represent snow shelters in the Blackburn National Forest. These snow shelters are illustrated in the following coordinate axes.

Horizontal scale: 1 unit represents 1 km.

Vertical scale: 1 unit represents 1 km.

The Park Ranger draws three straight lines to form an incomplete Voronoi diagram.

Calculate the gradient of the line segment AE.

[2]

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

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

[3]

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

[1]

Markscheme

$\frac{{3 - 1}}{{7 - 3}}$ (M1)

= 0.5 A1

[2 marks]

$y - 2 = - 2\left( {x - 5} \right)$ (A1) (M1)

Note: Award (A1) for their −2 seen, award (M1) for the correct substitution of (5, 2) and their normal gradient in equation of a line.

$2x + y - 12 = 0$ A1

[3 marks]

every point in the cell is closer to E than any other snow shelter A1

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.5—Intersection of lines, equations of perpendicular bisectors

22M.3.AHL.TZ1.2c.i:
Find the equation of the perpendicular bisector of $[IF]$ .
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.
22M.3.AHL.TZ1.2c.ii:
Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$ , find the coordinates of the other two vertices within the metropolitan area.
21N.1.SL.TZ0.7b.i:
Write down the equation of the perpendicular bisector of $[PC]$ .
22M.2.SL.TZ2.3a:
Find the midpoint of $[BD]$ .
22M.1.SL.TZ1.4a:
Find the equation of the line that the road follows.
22M.1.SL.TZ1.4b:
Town $C$ is due north of town $A$ and the road passes through town $C$ .

Find the $y$ -coordinate of town $C$ .
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.
SPM.1.SL.TZ0.7c:
In the context of the question, explain the significance of the Voronoi cell containing site E.
SPM.1.SL.TZ0.7b:
Find the equation of the line which would complete the Voronoi cell containing site E.

Give your answer in the form $ax + by + d = 0$ where $a$ , $b$ , $d \in \mathbb{Z}$ .
21M.1.SL.TZ2.6a:
Find the equation of the perpendicular bisector of $[AB]$ . Give your equation in the form $y = m x + c$ .
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.

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

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

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