Two bus stops \(P\) and \(Q\) lie 40 m apart along a straight road, shown in the diagram. A new stop must be nearer to \(P\) than to \(Q\). Two candidate si...
Question 1 Report
Two bus stops \(P\) and \(Q\) lie 40 m apart along a straight road, shown in the diagram. A new stop must be nearer to \(P\) than to \(Q\). Two candidate sites are marked: site 1, 15 m from \(P\), and site 2, 28 m from \(P\), both measured towards \(Q\).
- State the distance from \(P\) of the boundary line separating points nearer to \(P\) from points nearer to \(Q\). (1)
- Determine which site meets the requirement of being nearer to \(P\) than to \(Q\), giving a reason. (2)
Answer Details
The set of points equally distant from two fixed points is the perpendicular bisector of the segment joining them; any point closer to one of the two points than the other lies on that point's side of this boundary line.
(a) The boundary separating points nearer to \(P\) from points nearer to \(Q\) is the perpendicular bisector of \(PQ\), which lies at the midpoint of the 40 m distance between them:
\[40\div2=20 \text{ m from } P\] [1 mark](b) Site 1 is 15 m from \(P\). Since \(15 \lt 20\), site 1 lies on \(P\)'s side of the boundary, so it is nearer to \(P\) than to \(Q\), and it satisfies the requirement. Site 2 is 28 m from \(P\). Since \(28 \gt 20\), site 2 lies on \(Q\)'s side of the boundary, so it is nearer to \(Q\), and it does not satisfy the requirement. [2 marks]
Comparing each site's distance from \(P\) directly with the 20 m boundary found in part (a), rather than measuring from \(Q\) as well, is enough to decide which side of the perpendicular bisector each site falls on.
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