Problem Bank | IB Math AIHL | Perplex

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Three existing water towers are located at the points

A(0,0),B(8,0),C(0,6)

where each grid unit represents 1km. A new water tower is to be added to the diagram. To minimize risk of flooding, the water tower is placed at point P. A Voronoi diagram of the water towers is sketched below:

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Determine:
1. the coordinates of the point P that is equidistant from all 3 towers, and
  [3]
2. the common distance r from P to each tower.
  [2]

5

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A square park is modeled by the region 0≤x≤8 and 0≤y≤8 (all distances in meters).

There are three refreshment kiosks.

Kiosk A is at the point (2,6).
Kiosk B is at the point (2,b).
Kiosk C is at the point (6,6).

A Voronoi diagram of the park's kiosks is shown below:

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Every visitor is served by the nearest kiosk.

Show that V has coordinates (4,5).
[4]
Determine the value of b.
[3]
Determine the fraction of the park served by Kiosk A.
[2]

6

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A 6m×6m laboratory floor is the square 0≤x≤6, 0≤y≤6.

Four digital thermometers record the surface temperature (in °C) at the points:

Sensor	Coordinates	Recorded temperature
A	(1,1)	18°C
B	(5,1)	24°C
C	(5,5)	22°C
D	(1,5)	20°C

and a Voronoi diagram is sketched of the lab:

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Estimate the temperature at:
1. P(4,2);
  [6]
2. Q(3,3).
  [4]
Write down the equation of the edge between sensors B and C.
[2]

The laboratory manager estimates the temperature along the line x=4.

At point R, his measurements yield a result of 23°C.

Find the coordinates of R.
[2]