Loci

Key Facts and Exam Tips

• A locus is a set of points that obey a given rule. Examination questions will tell you the rule to follow in order to draw the set of points required. There are some standard loci (plural of locus) that you should know.
  
• The set of points at a constant distance from a fixed point is a circle.
  
• The set of points at a constant distance from a fixed line is a pair of lines parallel to the fixed line, plus a semi-circle at either end of the lines.
  
• The set of points equidistant (the same distance) from two parallel lines is a parallel line halfway between the two given lines.
  
• The set of points equidistant from two lines meeting at a point is a line halfway between the given lines. This line splits the angle where the lines meet in half. It is called the angle bisector, and you need to learn how to construct this using compasses.
  
• If the two given lines cross, then the set of points equidistant from the lines is a pair of lines that bisect the angles between the two given lines.
  
• The set of points equidistant from two given points is a line halfway between the points, perpendicular to (at right angles to) the line joining the given points. This is called the perpendicular bisector and you need to learn how to construct this, using compasses.
  
• Always use a sharp pencil, a ruler and compasses for construction work. Make any measurements as accurately as you can. Always use compasses to construct circles or parts of circles and a ruler to draw lines. Freehand sketches are not sufficient for examination answers.
  
• If a scale is used in the question, check that you have used the same scale accurately in your solution.
  
• You should leave all construction marks in your answer so that the examiner can see the steps you have taken. Compass arcs should be clearly visible but slightly fainter than the final locus that you have found.
  
• In some questions the locus can be a region, bounded by the set of points you have found. Decide on which side of the boundary the points that satisfy the given rule lie. Show the locus by shading the appropriate region. If the locus includes the boundary of the region, use a solid line to mark the boundary. If the boundary is not included it should be drawn as a dotted line.
  
• You can recognise if a region is required because the rule followed in the question will include an inequality (eg greater than, less than, more than, further than, at least, not more/less than, no greater/less than, greater than or equal to, etc).