Show that the new station should be built at $\text{P}$.

Write down the equation of the perpendicular bisector of $\text{[PC]}$.

Hence draw the missing boundaries of the region around $\text{P}$ on the following diagram.

(the best placement is either point $\text{P}$ or point $\text{Q}$)
attempt at using the distance formula              (M1)

$\text{AP}=\sqrt{{\left(10-6\right)}^2+{\left(6-2\right)}^2}$  OR

$\text{BP}=\sqrt{{\left(10-14\right)}^2+{\left(6-2\right)}^2}$  OR

$\text{DP}=\sqrt{{\left(10-10.8\right)}^2+{\left(6-11.6\right)}^2}$  OR

$\text{BQ}=\sqrt{{\left(13-14\right)}^2+{\left(7-2\right)}^2}$  OR

$\text{CQ}=\sqrt{{\left(13-18\right)}^2+{\left(7-6\right)}^2}$  OR

$\text{DQ}=\sqrt{{\left(13-10.8\right)}^2+{\left(7-11.6\right)}^2}$  

($\text{AP}$ or $\text{BP}$ or $\text{DP}$$=$)  $\sqrt{32}=5.66 \left(5.65685\dots \right)$  AND

($\text{BQ}$ or $\text{CQ}$ or $\text{DQ}$$=$)  $\sqrt{26}=5.10 \left(5.09901\dots \right)$            A1

$\sqrt{32}>\sqrt{26}$  OR  $\text{AP}$ (or $\text{BP}$ or $\text{DP}$) is greater than $\text{BQ}$ (or $\text{CQ}$ or $\text{DQ}$)            A1

point $\text{P}$ is the furthest away            AG

Note: Follow through from their values provided their $\text{AP}$ (or $\text{BP}$ or $\text{DP}$) is greater than their $\text{BQ}$ (or $\text{CQ}$ or $\text{DQ}$).

[3 marks]

$x=14$           A1

[1 mark]

           A1A1

Note: Award A1 for each correct straight line. Do not FT from their part (b)(i).

[1 mark]

In part (a) many candidates realized that distances were required. Many candidates seemed to have an idea about Voronoi diagrams. However, several candidates did not realize that they had to consider point $\mathrm{Q}$ as well in their comparison. Hence, several candidates only calculated distances from $\mathrm{P}$. The numerical comparison of the distance from $\mathrm{P} \left(\mathrm{AP}/\mathrm{BP}/\mathrm{DP}\right)$ and from $\mathrm{Q} \left(\mathrm{BQ}/\mathrm{CQ}/\mathrm{DQ}\right)$ need to be clearly shown. It was a pity to see that some candidates lost marks due to incorrect rounding of the values to three significant figures. The most common error being $5.09$. In part (b)(i) not many candidates seemed to understand what was required. A significant number of candidates wrote down the equation of the line through $\mathrm{PC}$, $y=6$, rather than the required line. In part (b)(ii), it seemed that much time was lost as many candidates attempted to find the equation of the perpendicular bisector of $\mathrm{DP}$ to draw the boundaries.

In part (a) many candidates realized that distances were required. Many candidates seemed to have an idea about Voronoi diagrams. However, several candidates did not realize that they had to consider point $\mathrm{Q}$ as well in their comparison. Hence, several candidates only calculated distances from $\mathrm{P}$. The numerical comparison of the distance from $\mathrm{P} \left(\mathrm{AP}/\mathrm{BP}/\mathrm{DP}\right)$ and from $\mathrm{Q} \left(\mathrm{BQ}/\mathrm{CQ}/\mathrm{DQ}\right)$ need to be clearly shown. It was a pity to see that some candidates lost marks due to incorrect rounding of the values to three significant figures. The most common error being $5.09$. In part (b)(i) not many candidates seemed to understand what was required. A significant number of candidates wrote down the equation of the line through $\mathrm{PC}$, $y=6$, rather than the required line. In part (b)(ii), it seemed that much time was lost as many candidates attempted to find the equation of the perpendicular bisector of $\mathrm{DP}$ to draw the boundaries.

In part (a) many candidates realized that distances were required. Many candidates seemed to have an idea about Voronoi diagrams. However, several candidates did not realize that they had to consider point $\mathrm{Q}$ as well in their comparison. Hence, several candidates only calculated distances from $\mathrm{P}$. The numerical comparison of the distance from $\mathrm{P} \left(\mathrm{AP}/\mathrm{BP}/\mathrm{DP}\right)$ and from $\mathrm{Q} \left(\mathrm{BQ}/\mathrm{CQ}/\mathrm{DQ}\right)$ need to be clearly shown. It was a pity to see that some candidates lost marks due to incorrect rounding of the values to three significant figures. The most common error being $5.09$. In part (b)(i) not many candidates seemed to understand what was required. A significant number of candidates wrote down the equation of the line through $\mathrm{PC}$, $y=6$, rather than the required line. In part (b)(ii), it seemed that much time was lost as many candidates attempted to find the equation of the perpendicular bisector of $\mathrm{DP}$ to draw the boundaries.