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Date	May 2021	Marks available	5	Reference code	21M.1.SL.TZ2.6
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Find	Question number	6	Adapted from	N/A

Question

Two schools are represented by points $A (2, 20)$ $A (2, 20)$ and $B (14, 24)$ $B (14, 24)$ on the graph below. A road, represented by the line $R$ $R$ with equation $- x + y = 4$ $- x + y = 4$ , 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.

Find the equation of the perpendicular bisector of $[AB]$ $[AB]$ . Give your equation in the form $y = m x + c$ $y = m x + c$ .

[5]

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

[2]

Markscheme

gradient $AB = \frac{4}{12} (\frac{1}{3})$ $AB = \frac{4}{12} (\frac{1}{3})$ (A1)

midpoint $AB : (8, 22)$ $AB : (8, 22)$ (A1)

gradient of bisector $= - \frac{1}{gradient AB} = - 3$ $= - \frac{1}{gradient AB} = - 3$ (M1)

perpendicular bisector: $22 = - 3 \times 8 + b$ $22 = - 3 \times 8 + b$ OR $(y - 22) = - 3 (x - 8)$ $(y - 22) = - 3 (x - 8)$ (M1)

perpendicular bisector: $y = - 3 x + 46$ $y = - 3 x + 46$ A1

[5 marks]

attempt to solve simultaneous equations (M1)

$x + 4 = - 3 x + 46$ $x + 4 = - 3 x + 46$

$(10.5, 14.5)$ $(10.5, 14.5)$ A1

[2 marks]

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 $[IF]$ $[IF]$ .
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 $(20, 37.5)$ $(20, 37.5)$ , 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 $[PC]$ $[PC]$ .
22M.2.SL.TZ2.3a:
Find the midpoint of $[BD]$ $[BD]$ .
22M.1.SL.TZ1.4a:
Find the equation of the line that the road follows.
22M.1.SL.TZ1.4b:
Town $C$ $C$ is due north of town $A$ $A$ and the road passes through town $C$ $C$ .

Find the $y$ $y$ -coordinate of town $C$ $C$ .
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.TZ0.7a:
Calculate the gradient of the line segment AE.
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 $a$ $a$ , $b$ $b$ , $d \in \mathbb{Z}$ .
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

Question

Markscheme

Examiners report

Syllabus sections

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