1. The Voronoi diagram below shows three identical cellular phone towers, T1, T2 and T3. A fourth identical cellular phone tower, T4 is located in the shaded region. The dashed lines in the diagram below represent the edges in the Voronoi diagram. Horizontal scale: 1 unit represents 1 km. Vertical scale: 1 unit represents 1 km. -15 Tim stands inside the shaded region. T3 10- -2- a. Explain why Tim will receive the strongest signal from tower T4. [1 mark] Tower T2 has coordinates (-9, 5) and the edge connecting vertices A and B has equation y = 3. b. Write down the coordinates of tower T4. [1 mark] Tower T1 has coordinates (-13, 3). c. Find the gradient of the edge of the Voronoi diagram between towers T1 and T2. [3 marks] 2. There are four stations used by the fire wardens in a national forest. On the following Voronoi diagram, the coordinates of the stations are A(6, 2), B(14, 2), C(18, 6) and D(10.8, 11.6) where distances are measured in kilometres. The dotted lines represent the boundaries of the regions patrolled by the fire warden at each station. The boundaries meet at P(10, 6) and Q(13, 7). 16- 14- 12- D' 10 10 8 6- 9 4- 2. A P a B to -+ 6 -60 8 10 12 14 16 18 To reduce the areas of the regions that the fire wardens patrol, a new station is to be built within the quadrilateral ABCD. The new station will be located so that it is as far as possible from the nearest existing station. a. Show that the new station should be built at P. [3 marks] The Voronoi diagram is to be updated to include the region around the new station at P. The edges defined by the perpendicular bisectors of [AP] and [BP] have been added to the following diagram. b. Write down the equation of the perpendicular bisector of [PC]. [1 mark] 16- 14- 12- 10 8- 6 4 2. A D' B to 2 4 6 8 10 12 14 16 18 c. Hence draw the missing boundaries of the region around P on the following diagram. [2 marks] 16- 14- 12- D' 10- 8 60 4 2 0 ~- 2 4 A -6 P B C -00 8 10 12 14 16 18 x