Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Constructions and loci. 5 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Constructions are accurate drawings made with a ruler and compasses, and a locus is the set of points that follow a rule. You construct bisectors, perpendiculars and triangles, and shade regions that satisfy several conditions, leaving all your construction arcs showing.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Construct triangles from given sides and anglesUse a ruler and protractor for SAS and ASA, and a ruler and compasses for SSS.
4
Construct the perpendicular bisector of a lineDraw arcs of equal radius from each end, then join the two points where they cross.
4
Construct the bisector of an angleDraw an arc from the vertex, then equal arcs from where it crosses the arms, and join to the vertex.
5
Construct perpendiculars from and at a pointConstruct the perpendicular from a point to a line, or at a point on a line, with compasses.
5
Draw loci and shade the required regionCombine loci such as 'within 3 cm of A' and 'closer to B than to C', then shade where all are true.
5
Solve loci problems on scale drawingse.g. show where a tree can be planted in a garden, using a scale drawing and constructions.
Notes
Standard constructions
Perpendicular bisector of AB: open the compasses to more than half of AB. Draw arcs from A and from B, with the same radius, above and below the line. Join the two points where the arcs cross.
Angle bisector: draw an arc centred on the vertex, crossing both arms. From each crossing point, draw arcs of the same radius that meet. Join the vertex to where they meet.
Perpendicular from a point P to a line: draw an arc centred on P that crosses the line twice. From these two points, draw equal arcs that meet on the other side of the line. Join P to where they meet.
Perpendicular at a point P on a line: draw arcs centred on P crossing the line on both sides, then construct the perpendicular bisector of those two points.
The perpendicular distance from a point to a line is the shortest distance from the point to the line.
An equilateral triangle gives 60°; bisecting it gives 30°. Bisecting a right angle gives 45°.
Constructing triangles
SSS: draw one side with a ruler. Set the compasses to each of the other two lengths and draw arcs from the ends of that side. Join both ends to where the arcs cross.
SAS and ASA: use a ruler for the sides and a protractor for the angles.
Loci
A locus is the set of all points that obey a rule.
A fixed distance from a point: a circle.
A fixed distance from a line segment: two parallel lines joined by a semicircle at each end.
Equidistant from two points: the perpendicular bisector. Equidistant from two lines: the angle bisector.
Closer to A than to B: the side of the perpendicular bisector that contains A. Less than 3 cm from a point: inside a 3 cm circle.
With several conditions, draw each locus, then shade the region where all of them are true.
Cheatsheet
Equidistant from two points → perpendicular bisector
Equidistant from two lines → angle bisector
Fixed distance from a point → circle
Fixed distance from a line segment → two parallel lines with semicircular ends
Shortest distance from a point to a line → the perpendicular
Use ruler and compasses only to construct a bisector
2 marks4
Set the compasses to more than half the length of the line (or any suitable radius for an angle).
Draw both pairs of arcs without changing the compass width.
Draw the bisector through the crossing points, leaving all arcs visible.
Example. Draw a line AB 8 cm long. Use ruler and compasses only to construct the perpendicular bisector of AB. You must show all your construction lines.
Show the model answer
Arcs of the same radius (more than 4 cm), centred on A and on B, crossing above and below AB (M1) A straight line through both crossing points, meeting AB at right angles 4 cm from A, with the arcs left showing (A1)
Construct a triangle
2 marks3
Draw the longest side first with a ruler.
Draw an arc from each end with the compasses set to the other two lengths.
Join each end to the point where the arcs cross.
Example. Use ruler and compasses to construct triangle PQR with PQ = 7 cm, QR = 5 cm and PR = 6 cm. You must show all your construction lines.
Show the model answer
PQ drawn 7 cm long, with an arc of radius 6 cm centred on P and an arc of radius 5 cm centred on Q (M1) R at the point where the arcs cross, joined to P and Q, all sides accurate to within 2 mm (A1)
Loci: show the region by shading
3 marks5
Turn each condition into a locus: a circle, a perpendicular bisector or an angle bisector.
Construct each locus accurately on the diagram (using the scale).
Shade only the region where every condition is true, and label it.
Example. A rectangular garden ABCD has AB = 10 m and BC = 6 m. A scale drawing uses 1 cm to represent 1 m. A tree is to be planted closer to AB than to AD, and less than 4 m from C. On a scale drawing, show by shading the region where the tree can be planted.
Show the model answer
Rectangle drawn 10 cm by 6 cm; bisector of angle DAB constructed with compasses (M1) Arc of radius 4 cm centred on C (M1) Region shaded inside the garden, inside the arc and on the AB side of the angle bisector (A1)
Shortcuts and memory tricks
Keep the compass width the same for each pair of arcs.
Check a perpendicular bisector afterwards: it should cut the line exactly in half at 90°.
'Closer to' means a bisector; 'within' or 'less than' a distance from a point means inside a circle; 'at least' means outside it.
Test one point in your shaded region against every condition before you finish.
Where marks are lost
Rubbing out construction arcs; without them the construction scores no marks.
Using a protractor when the question says 'ruler and compasses only'.
Setting the compasses to less than half the line, so the arcs do not cross.
Drawing a full circle when the locus is limited by a boundary such as a wall or the edge of a garden.
Shading the wrong side of a bisector.
Exam technique
'Use ruler and compasses' means every construction arc must be visible in your answer.
Use a sharp pencil and compasses that do not slip; accurate work is only allowed an error of about 2 mm or 2°.
Read each condition carefully: 'less than', 'at least', 'closer to' and 'equidistant from' mean different regions.
Label the required region (e.g. with R) as well as shading it.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
The diagram shows the line segment AB. This question should be answered on a printed copy of the diagram.
[object Object]
Use ruler and compasses to construct the perpendicular bisector of AB. You must show all your construction lines.[2]
Show the answer and mark scheme
Answer: A straight line through the two points where arcs of equal radius, centred on A and on B, intersect (one point on each side of AB).
M1 for two pairs of intersecting arcs of equal radius, centred on A and on B, one pair on each side of AB
A1 for a correct perpendicular bisector drawn through the points of intersection of the arcs (within 2 mm of the correct line)
Worked solution: Open the compasses to more than half of AB. Draw arcs centred on A above and below AB. With the same radius, draw arcs centred on B to cross the first arcs. Join the two crossing points with a straight line: this line is the perpendicular bisector of AB.
Question 2Medium3 marks
This question should be answered on plain paper using a ruler, compasses and a protractor.
(a) Use ruler and compasses to construct triangle ABC with AB = 8 cm, AC = 6 cm and BC = 7 cm. You must show all your construction lines.[2]
(b) Measure the size of angle ABC.[1]
Show the answer and mark scheme
(a)Answer: AB drawn 8 cm; arc radius 6 cm centred on A and arc radius 7 cm centred on B, crossing at C.
M1 for AB drawn 8 cm long and an arc of radius 6 cm centred on A or an arc of radius 7 cm centred on B
A1 for a correct triangle with both construction arcs shown (sides within 2 mm)
Worked solution: Draw AB = 8 cm. Set the compasses to 6 cm and draw an arc centred on A. Set the compasses to 7 cm and draw an arc centred on B. The arcs cross at C. Join AC and BC.
(b)Answer: 47° (allow 45° to 49°) °
B1 for an angle in the range 45° to 49°
Worked solution: cos ABC = (8² + 7² − 6²) ÷ (2 × 8 × 7) = 77 ÷ 112, so angle ABC ≈ 46.6°. A careful construction measures about 47°.