IGCSE Mathematics: Geometrical Terms and Constructions Practice Questions
Constructions must be done with compasses and a straight edge, leaving all arcs visible. A locus is the set of all points satisfying a condition, and the two most common are the perpendicular bisector and the angle bisector.
Topic 4.1 of Cambridge IGCSE Mathematics 0580 awards marks for the construction arcs, not just the final line. Erasing your working guarantees losing marks. The questions below cover both constructions, loci regions and bearings.
What you need to know for Geometrical Terms and Constructions
- Perpendicular bisectorThe locus of points equidistant from two fixed points. Construct it with equal arcs from both points, above and below the line.
- Angle bisectorThe locus of points equidistant from two lines. Construct it with an arc from the vertex, then equal arcs from where it crosses each arm.
- Locus from a pointPoints a fixed distance from a point form a circle. Points less than that distance form the region inside it.
- Locus from a linePoints a fixed distance from a straight line form two parallel lines, joined by semicircles at the ends if the line is a segment.
- BearingsMeasured clockwise from north and always written with three figures, such as 072 degrees.
- Back bearingThe reverse bearing differs by 180 degrees. Add 180 if the original is below 180, and subtract 180 if it is above.
IGCSE Mathematics Geometrical Terms and Constructions questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Describe how to construct the perpendicular bisector of a line segment AB using only compasses and a straight edge.
Show the worked answer
- Open the compasses to a radius greater than half the length of AB. The exact size does not matter as long as the arcs will cross.
- Place the compass point on A and draw arcs above and below the line.
- Without changing the radius, place the compass point on B and draw arcs that cross the first pair, above and below the line.
- Join the two points where the arcs cross with a straight line. That line is the perpendicular bisector, and every point on it is equidistant from A and B. Leave all construction arcs visible.
Describe how to construct the bisector of an angle using compasses and a straight edge.
Show the worked answer
- Place the compass point on the vertex of the angle and draw an arc that crosses both arms.
- Place the compass point where the arc crosses the first arm and draw an arc inside the angle.
- Without changing the radius, repeat from where the arc crosses the second arm, so the two new arcs cross.
- Draw a straight line from the vertex through that crossing point. Every point on this line is equidistant from the two arms of the angle.
A goat is tethered by a 4 m rope to a post in the middle of a large field. Describe the locus of points the goat can reach, and describe how the region would change if the goat were tethered to a rail rather than a post.
Show the worked answer
- Tethered to a post, the goat can reach any point up to 4 m from the post in every direction.
- The boundary is a circle of radius 4 m centred on the post, and the region it can reach is the area inside that circle.
- Tethered to a straight rail, the goat can reach any point within 4 m of any part of the rail.
- That region is a rectangle 4 m either side of the rail, with a semicircle of radius 4 m at each end where the rope pivots around the rail's ends.
The bearing of B from A is 070 degrees. Calculate the bearing of A from B.
Show the worked answer
- A back bearing differs from the original by 180 degrees.
- The original bearing is 070 degrees, which is less than 180, so add 180.
- 070 + 180 = 250 degrees.
- The bearing of A from B is 250 degrees. Bearings are always written with three figures, so 070 rather than 70.
Common mistakes in this topic
- Erasing construction arcs, which removes the evidence the marks are awarded for.
- Measuring with a ruler or protractor when a construction is required.
- Changing the compass radius partway through a construction.
- Writing a bearing with fewer than three figures.
- Drawing a locus boundary without shading or labelling the required region.
Exam tips
- Never rub out construction arcs. They are what the marks are for.
- Use a sharp pencil and keep the compasses tight so the radius does not slip.
- Read whether the question wants a boundary line or a shaded region, and label which is which.
- Bearings are always three figures, measured clockwise from north.
- For combined loci, draw each condition separately, then shade only the overlap.
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Geometrical Terms and Constructions FAQs
How do I construct a perpendicular bisector?
Open the compasses to more than half the length of the line, draw arcs above and below from each endpoint using the same radius, then join the two points where the arcs cross. Leave all arcs visible, because marks are awarded for the construction as well as the line.
What is a locus?
A locus is the set of all points satisfying a given condition. Points a fixed distance from a point form a circle, points equidistant from two points form a perpendicular bisector, and points equidistant from two lines form an angle bisector.
How do I find a back bearing?
Add or subtract 180 degrees. If the original bearing is less than 180, add 180. If it is 180 or more, subtract 180. So the bearing of A from B, where B from A is 070 degrees, is 250 degrees.
Why do I lose marks for measuring instead of constructing?
A construction question tests the compass method specifically, and the marks are allocated to the arcs. A line drawn accurately with a ruler and protractor shows none of that method, so it typically scores nothing even when the position is correct.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.