Sheet № 60 · Foundation + Higher · AQA · Edexcel · OCR
Constructions and Loci –
Constructions and loci is a practical geometry topic tested at both Foundation and Higher tiers on AQA, Edexcel, and OCR GCSE Maths papers. You need to know how to use a pair of compasses and a straight edge (ruler) to construct perpendicular bisectors, angle bisectors, and other standard constructions accurately. You also need to underst
§Key definitions
Question:
Two mobile phone masts are at points A and B, which are 8 cm apart on a map. The signal from mast A reaches up to 5 cm and the signal from mast B reaches up to 4 cm. Shade the region that receives signal from both masts.
Answer:
The shaded region is the intersection of the two circles.
Question 1:
Construct the perpendicular bisector of a line segment 7 cm long.
Question 2:
Construct the bisector of a 70° angle.
Question 3:
Draw the locus of points exactly 4 cm from point P.
§Formulas to memorise
Fixed distance from a point — a circle.
Fixed distance from a line — a "racetrack" shape (two parallel lines with semicircular ends).
Equidistant from two points — the perpendicular bisector of the line segment joining the two points.
Equidistant from two lines — the angle bisector of the angle formed by the two lines.
This line bisects the angle — it divides it into two equal parts.
Worked example
Two mobile phone masts are at points A and B, which are 8 cm apart on a map. The signal from mast A reaches up to 5 cm and the signal from mast B reaches up to 4 cm. Shade the region that receives signal from both masts.
Working:
⚠ Common mistakes
- ✗Rubbing out construction arcs. This is the most common error. Leave all arcs visible — they prove you used the correct construction method.
- ✗Setting the compass width too small. For a perpendicular bisector, the compass must be set to more than half the line length, otherwise the arcs will not intersect.
- ✗Inaccurate compass work. Make sure the compass point does not slip. Hold it firmly at the centre and rotate smoothly.
- ✗Confusing "closer to" with "equidistant from". "Equidistant" means on the bisector line. "Closer to A than B" means on A's side of the perpendicular bisector.
- ✗Not reading all conditions. Loci questions often have two or three constraints. Shade only the region satisfying all of them simultaneously.
✦ Exam tips
- →Bring a sharp pencil and a reliable pair of compasses. Accuracy matters — marks can be lost if constructions are more than 2 mm out.
- →Label your constructions. If the question asks for a perpendicular bisector, write "perpendicular bisector" next to it.
- →For combined loci questions, draw each locus separately first (in pencil), then identify and shade the region satisfying all conditions.
- →Use a ruler to check distances after drawing, to verify your construction is accurate.
- →Practise with real equipment. This is one of the few topics where physical practice (as opposed to just doing calculations) makes a significant difference.