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Core + Extended Geometry

Constructions and Loci for IGCSE Maths

Using ruler and compasses for constructions and loci. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding constructions and

What You Need to Know

Using ruler and compasses for constructions and loci. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding constructions and loci is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Constructions and Loci

Geometrical constructions use a ruler and compasses only (no protractor). Key constructions in IGCSE 0580: perpendicular bisector of a line segment, bisector of an angle, 60° angle (equilateral triangle method), and perpendicular from/to a line. A locus is the set of all points satisfying a condition. Key loci: equidistant from two points (perpendicular bisector), fixed distance from a point (circle), equidistant from two lines (angle bisector), fixed distance from a line (parallel lines).

Step-by-Step Method

  1. 1

    Perpendicular bisector of AB

    Open compasses to more than half AB. From A and B, draw arcs above and below the line. The arcs intersect at two points — join them.

  2. 2

    Angle bisector of angle ABC

    With centre B, draw an arc cutting BA and BC. From each arc intersection, draw equal arcs to meet inside. Join B to this meeting point.

  3. 3

    Draw a 60° angle

    Draw a baseline. From a point, draw an arc. From where the arc cuts the baseline, draw another equal arc. Join the intersections for 60°.

  4. 4

    Locus: equidistant from two lines

    Use the angle bisector construction — the bisector is the locus of points equidistant from both lines.

  5. 5

    Shade the required locus region

    For combined loci, construct each boundary, then shade (or leave unshaded) the region satisfying all conditions.

Worked Example

Question

Construct the locus of points inside triangle ABC that are (i) closer to AB than to BC, and (ii) within 3 cm of point A.

Solution

Step 1: Construct the angle bisector of angle B (equidistant from AB and BC). Points on the AB side of this bisector are closer to AB. Step 2: Draw a circle of radius 3 cm centred at A. Step 3: The required region is inside the triangle, on the AB side of the angle bisector, AND inside the circle of radius 3 centred at A. Shade or label the overlap of these two regions inside triangle ABC.

Exam Tips for Constructions and Loci

  • Leave all construction arcs visible — do not erase them. The examiner needs to see them to award marks.
  • Use a sharp pencil and draw lightly — precision matters in construction questions.
  • The perpendicular bisector is the locus of points equidistant from two points — state this connection explicitly.
  • The angle bisector is the locus of points equidistant from two lines — state this when describing loci.

Practice Questions

Q1: Describe the locus of a point that is always 5 cm from a fixed point O.

Show hint

A circle of radius 5 cm centred at O.

Q2: Describe the locus of a point equidistant from two parallel lines l₁ and l₂.

Show hint

A line parallel to both l₁ and l₂, halfway between them.

Q3: Construct the perpendicular bisector of a 6 cm line segment. What is the locus of points equidistant from the two endpoints?

Show hint

The perpendicular bisector IS the locus of all points equidistant from both endpoints.

Frequently Asked Questions

What is constructions and loci in IGCSE Maths?

Using ruler and compasses for constructions and loci.

Is constructions and loci in the Core or Extended syllabus?

Constructions and Loci is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise constructions and loci effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on constructions and loci rather than trying to cover everything at once.

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