Constructions & Loci (AQA GCSE Maths)

What are constructions?

• A construction is a process where you create particular geometric objects using only a pair of compasses and a straight edge
• It is an excuse to get the maths toys out – rulers and compasses in particular!
• You will often be working with scale drawings with this topic

1. Perpendicular bisector

• This is a line that cuts another one exactly in half (bisect) but also crosses it at a right angle (perpendicular)
• It shows a path that is equidistant (equal distance) between two places
  • If you are on one side or the other of this line you know you are closer to one place than the other
• To construct a perpendicular bisector, you will need a sharp pencil, ruler and set of compasses
  • You may be asked to construct a perpendicular bisector of a line segment or to find the region that is closer to one point than another, the following method is used for both
  • STEP 1

    Set the distance between the point of the compasses and the pencil to be more than half the length of the line

  • STEP 2

    Place the point of the compasses on one end of the line and sketch an arc above and below the line

  • STEP 3

    Keeping your compasses set to the same distance, move the point of the compasses to the other end of the line and sketch an arc above and below it again, the arcs should intersect each other both above and below the line

  • STEP 4

    Connect the points where the arcs intersect with a straight line

• Final result:

2. Perpendicular to a line, from a point

• This is drawing a line that is perpendicular to another one but the line has to go through a particular point
  • It is essentially the same as the perpendicular bisector with one extra step to do first
• They show the shortest distance from the point to a line
  • eg. the shortest distance from a lighthouse to the coastline

• To construct a perpendicular from a point to a line, you will need a sharp pencil, ruler and set of compasses
  • You may be asked to construct a perpendicular from a point to a line or to find the shortest distance from a point to a line, the following method is used for both

• • STEP 1

    Set the distance between the point of your compasses and the pencil to be greater than the distance between the point P and the line

  • STEP 2

    Placing the point of the compasses onto the point P, draw an arc that intersects the line in two places

  • STEP 3

    Make sure that the distance between the point on the compasses and the pencil is greater than half the distance between the two points of intersection, place the point of the compasses on to one point of intersection and sketch an arc above and below the line

  • STEP 4

    Keeping the distance between the point on the compasses and the pencil the same, move th point of the compasses onto the other point of intersection and again draw an arc both above and below the line, these should intersect with the previous arcs

  • STEP 5
    Join the points where the arcs intersect each other and the original point P with a straight line

• Final result:

3. Angle bisector

• This is a line that cuts an angle exactly in half (bisect),
  • Either side of the line shows you are closer to one of the angle lines than the other
  • It is essentially another variation on the perpendicular bisector

• It shows a region on a map/diagram that is closer to one side than another

• To construct an angle bisector, you will need a sharp pencil, ruler and set of compasses
  • You may be asked to construct a bisector of an angle or to find the region that is closer to one side of an angle than the other, the following method is used for both
  • STEP 1

    Open the compasses, the distance between the point and the pencil is not particularly important but setting them about half the distance of the lines forming the angle is reasonable

  • STEP 2

    Placing the point of the compasses at the point of the angle, sketch an arc that intersects both of the lines that form the angle

  • STEP 3

    Set your compasses to the distance between these two points of intersection, place the point of the compasses on one of the points of intersection and sketch an arc

  • STEP 4

    Keeping the distance between the point of the compasses and the pencil the same, place the point of the compasses on the other point of intersection and sketch an arc, this should intersect the arc sketched in STEP 3

  • STEP 5
    Join the point of the angle to the point of intersection with a straight line

• Final result:

What are Loci?

• A locus (loci is plural) is a line/shape/path that is determined by following a rule 
  • eg. always being 2 m away from a wall
• You may be asked to construct a locus, although the language used in exam questions won’t always mention these words as questions are often based on real-world situations
• You may be expected to use some of the constructions mentioned in the revision note on constructions
• You may need to use a ruler, a protractor or a pair of compasses

What are the common types of loci?

• Points that are a fixed distance from a point
  • This will form a circle around the point
    • Use a pair of compasses
• Points that are a fixed distance from a line
  • This will form a race track shape
    • Two parallel lines
    • Two semi-circles at the end of the lines
• Points that are equidistant from two points
  • This will form the perpendicular bisector between the line segment between the two points
• Points that are equidistant from two lines
  • This will form the angle bisector of the angle between the two lines