Geometry · GCSE Maths
Geometric constructions
GCSE Maths constructions: perpendicular bisector, angle bisector, perpendicular from a point, 60° angles, and loci with compasses and ruler only.
Constructions need visible arcs. Bisector: arcs from both ends. Perpendicular at a point: arc across the line, then arcs from the crossings. 60°: equilateral triangle hack.
The important bits
What you need to know
- 1
Use a sharp pencil, ruler and compasses only unless the question allows a protractor. Construction lines and arcs must be visible.
- 2
Perpendicular bisector of AB: arcs from A and B with the same radius (> half AB), join the two arc intersections.
- 3
Angle bisector: arc from the vertex crossing both arms; arcs from those crossings with same radius; join vertex to intersection.
- 4
Perpendicular from a point on a line: from the point, arc across the line; from crossings, arcs that meet above (or below) the line.
- 5
Perpendicular from a point not on the line: arc centred on the point crossing the line twice; bisect that chord on the line.
- 6
60° angle at a point: draw a base line, arc centre at one end radius r, arc same radius from intersection making third point of equilateral triangle — angle 60°.
- 7
Loci: equidistant from A and B is the perpendicular bisector of AB. Fixed distance from P is a circle centre P.
- 8
Do not rub out construction arcs if the question says “show all construction lines” — they are part of the method.
Quotations worth analysing
Short evidence. Real method.
“Join the points where the arcs cross”
Those two crossings are equidistant from A and B because the arcs have equal radius from each end. The line is the locus.
“Same radius from both arms”
Equal-radius arcs from the two points on the arms are equidistant from the arms. The bisector is the set of points equidistant from both arms.
“Arcs must be visible for construction marks”
A correct final line with no arcs scores like a guess. Freehand perpendiculars fail construction criteria.
Go deeper
Perpendicular bisector in context
Construct the perpendicular bisector of PQ. Open compasses more than half PQ. Arc from P, arc from Q same radius, intersections above and below PQ. Join intersections — infinite line bisecting PQ at right angles. To find a point equidistant from P and Q on a diagram, it lies on this line. Region closer to P than Q is on the P side — half-plane. Exam: construct and shade the region closer to A than B and within 5 cm of C — bisector plus circle locus, shade overlap.
Go deeper
Angle bisector and 60°
Bisect angle ABC. Arc from B crossing BA and BC. From those two points, equal arcs (same radius) inside the angle. Join B to where those arcs meet. Every point on the line is equidistant from BA and BC. For 60° at A on line AB: arc centre A radius r meeting line at D. Arc centre D radius r meeting first arc at E. Triangle ADE equilateral, angle DAE = 60°. Join A to E for the 60° ray. 90°: bisect a straight angle or construct perpendicular. 45°: bisect 90°.
Go deeper
Perpendicular from a point
Point P on line L: small arc from P crossing L at two points. From those, two arcs (equal radius) meeting on one side. Join P to meeting — perpendicular. Point Q not on L: wide arc from Q crossing L at R and S. Bisect RS (chord on L) — that bisector passes through Q and is perpendicular to L (locus of centres through chord). GCSE combines: construct triangle with AB = 6 cm, angle 60° at A, perpendicular bisector of AB to find equidistant point from A and B. Show all arcs.
See the idea in action
Construct the perpendicular bisector of a 8 cm line AB and mark the midpoint. Step 1: Draw AB = 8 cm. Step 2: Open compasses to more than 4 cm. Arc from A above and below AB. Step 3: Same radius, arcs from B above and below. Two pairs of intersections. Step 4: Join the upper pair (or lower pair) — perpendicular bisector. Step 5: Bisector crosses AB at midpoint M, 4 cm from A and B. Leave arcs visible.
Exam technique
Turn knowledge into marks
Read “construct” as compasses required. Leave construction arcs. Label the final line or angle. Check with a set square only after drawing — not instead of construction.
Common mistakes
Do not give these marks away
- 01
Using a protractor when the question requires a construction.
- 02
Rubbing out arcs and losing construction marks.
- 03
Radius too small on perpendicular bisector — arcs do not cross.
To construct a perpendicular bisector of AB, the compasses radius must be
ALess than half AB
BMore than half AB
CExactly half AB
DEqual to AB only
Show the answer
More than half AB. Arcs from A and B must intersect. If radius ≤ half AB, arcs do not cross. Exactly half can touch at one point — not two stable intersections.
Quick questions
If this is the bit you searched
Why must construction arcs stay visible?
They prove the method. Examiners award construction marks for correct arcs and joining, not only the final line.
How do I construct a 60° angle?
Use equilateral triangle construction: equal arcs from a point on the base give 60° at the vertex.
What is the difference between bisector constructions?
Perpendicular bisector of a line: equidistant from endpoints. Angle bisector from a vertex: equidistant from two arms.
Can I use a protractor for constructions?
Only if the question allows measuring. “Construct” without exception means compasses and ruler.