Constructions & Loci
Ruler-and-compass constructions, scale drawings, and loci - regions defined by distance from points and lines.
Learn
Constructions & Loci, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Construct a perpendicular bisector and an angle bisector with ruler and compasses
Perpendicular bisector: arcs of equal radius from each end of the line, crossing above and below - join the crossings. Angle bisector: an arc from the vertex cutting both arms, then equal arcs from those cuts - join the vertex to where they meet. Leave every arc visible; the arcs ARE the marks.
Construct triangles, and angles of 60° and 90°
For a triangle from three sides, draw the base, then arc each remaining length from its end - the apex is where the arcs cross. A 60° angle comes from one arc step around a circle (the equilateral construction); 90° comes from a perpendicular bisector on a straight line.
Draw loci: a fixed distance from a point or a line, and equidistant from two points or two lines
Fixed distance from a point: a circle. From a line segment: a running track - parallel lines with semicircular ends. Equidistant from two points: their perpendicular bisector; from two lines: the angle bisector. Every locus question is one of these four.
Shade regions that satisfy several conditions at once
Draw each condition's boundary - a circle for "within 3 m of the tap", a bisector for "closer to A than B" - then shade where the required sides overlap. Read carefully whether points must be nearer, further, inside or outside each boundary.
Interpret and use scale drawings
Convert real distances with the scale before setting your compasses: at 1 cm to 5 m, a 20 m rope is a 4 cm circle. Answers read from the drawing convert back the other way. Accuracy matters - measure to the millimetre.
Practice
Try a Constructions & Loci question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
A rectangular field measures 80 m by 60 m. A scale drawing uses 1 cm to 20 m. What is the area of the scale drawing in cm²?
- 4800 cm²
- 240 cm²
- 12 cm²
- 7 cm²
Show the answer and the working
Answer: 12 cm²
Convert each side to the drawing scale first, then multiply the drawing dimensions.
- Convert each side using 1 cm to 20 m: 80 ÷ 20 = 4 cm and 60 ÷ 20 = 3 cm.
- The drawing is a 4 cm by 3 cm rectangle.
- Area = 4 × 3 = 12 cm².