Step-by-step worked examples and graded practice questions on constructions, loci and bearings — perpendicular and angle bisectors, the four standard loci, and three-figure bearings.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
Struggling with constructions or bearings?
Alamin diagnoses exactly which skills are missing and builds a structured plan to fix them — backed by AI-powered practice between sessions.
A construction is a precise drawing made using only a straight edge and compasses — no protractor, no measuring the angle directly. Exam papers expect your construction lines (the compass arcs) to be left visible, not rubbed out.
Worked Example 1 — Perpendicular bisector
Construct the perpendicular bisector of the line segment AB.
1
Open your compasses to more than half the length of AB.
2
With the point on A, draw an arc above and below the line. Without changing the compass width, repeat with the point on B.
3
Draw a straight line through the two points where the arcs cross. This line is the perpendicular bisector — every point on it is equidistant from A and B.
The dashed arcs are the compass construction; the solid amber line is the perpendicular bisector of AB.
Worked Example 2 — Angle bisector
Construct the bisector of angle O.
1
With the point on O, draw an arc that crosses both lines of the angle.
2
From each of these two crossing points, draw a new arc of equal radius into the middle of the angle, so the two arcs intersect.
3
Draw a straight line from O through the point where the arcs cross. This line bisects the angle exactly in half.
The dashed arcs are the compass construction; the solid amber line bisects the angle at O.
Loci
A locus (plural: loci) is the set of all points that satisfy a given rule. There are four standard loci you need to know:
1. Equidistant from a fixed point
The locus of points a fixed distance from a point is a circle centred on that point.
The locus of points 3cm from A is a circle of radius 3cm.
2. Equidistant from two fixed points
The locus of points equidistant from two points A and B is the perpendicular bisector of AB.
The perpendicular bisector of AB splits the plane into "closer to A" and "closer to B".
3. A fixed distance from a line
The locus of points a fixed distance from a straight line segment is a "racetrack" shape — two lines parallel to the segment, joined by semicircles at each end.
The locus of points 3cm from segment PQ is a racetrack shape: parallel lines plus semicircles at each end.
4. Equidistant from two lines
The locus of points equidistant from two straight lines is the angle bisector between them.
The angle bisector from O is equidistant from both lines l₁ and l₂.
Bearings
A bearing gives a direction as an angle measured clockwise from North, always written using three figures (e.g. 060°, not 60°).
Worked Example 3 — Reading a bearing
Find the bearing of B from A.
1
Draw a North line at A (the point you're measuring from).
2
Measure the angle clockwise from North to the line AB.
AnswerBearing of B from A = 060°
The bearing of B from A is measured clockwise from North at A.
Worked Example 4 — Back bearings
The bearing of B from A is 060°. Find the bearing of A from B (the back bearing).
1
North lines at A and B are parallel, so co-interior angle facts apply.
2
If the bearing is less than 180°, add 180°. If it's 180° or more, subtract 180°.
3
060° + 180° = 240°
AnswerBearing of A from B = 240°
North lines at A and B are parallel, so the two bearings are related by ±180°.
Practice questions
Work through each question before checking the answers. Where a question shows a diagram, it's required to answer it.
Foundation (Grade 4–6)
Q1Describe the locus of points that are exactly 4cm from a fixed point A.Foundation
Q2Describe, step by step, how to construct the perpendicular bisector of a 6cm line segment.Foundation
Q3A ship sails from port on a bearing of 070°. What bearing must it sail on to return directly to port?Foundation
Q4The diagram shows the bearing of B from A. State the bearing.Foundation
Diagram for Question 4 (not to scale).
Q5Describe the locus of points equidistant from two fixed points C and D, 8cm apart.Foundation
Higher (Grade 5–7)
Q6A wall runs in a straight line. Describe the locus of points exactly 2m from the wall.Higher
Q7A goat is tied by a 3m rope to a post in the middle of a large flat field. Describe the region the goat can graze.Higher
Q8The bearing of a lighthouse L from a boat B is 048°. Find the bearing of B from L.Higher
Q9A garden is a rectangle. A tree is to be planted so that it is within 5m of corner A, and closer to side XY than to side YZ. Describe how you would construct the boundary of the possible planting region.Higher
Q10The bearing of X from Y is 128°. Find the bearing of Y from X.Higher
Higher — Hard (Grade 8–9)
Q11Describe how to construct an angle of 60° using only compasses and a straight edge (no protractor).Grade 8–9
Q12A boat sails from port on a bearing of 245° for 8km, then changes course to a bearing of 100° for 5km. Find the bearing the boat must sail on to return directly to port. (Hint: find the bearing of the return leg from the final position, using the second leg's bearing.)Grade 8–9
Q13Point B is on a bearing of 072° from A. Point C is on a bearing of 155° from A. Find the angle BAC.Grade 8–9
Q14A point P is 5cm from fixed point A, and also equidistant from fixed points B and C. Describe how you would find P using constructions, and state how many such points exist in general.Grade 8–9
Q15A is due west of B. C is on a bearing of 052° from B, and the bearing of C from A is 038°. Find angle ACB.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1A circle of radius 4cm, centred on A
Q2Open compasses to more than 3cm; draw arcs above and below the line from each end; join the two crossing points with a straight line
Q3250°(070° + 180°)
Q4115°
Q5The perpendicular bisector of CD
Higher (Q6–Q10)
Q6Two straight lines parallel to the wall, one on each side, both 2m away
Q7A circle of radius 3m, centred on the post
Q8228°(048° + 180°)
Q9Draw an arc of radius 5m centred on A (locus 1); construct the angle bisector of angle Y between sides XY and YZ (locus 2); the boundary of the planting region is where these overlap
Q10308°(128° + 180°)
Higher — Hard (Q11–Q15)
Q11Draw a line and an arc from one end; without changing the compass width, draw a second arc from where the first arc crosses the line; join the start point to where the two arcs cross — this constructs an equilateral triangle, so the angle is 60°
Q12100° − 180° = −80°, so 280°; the return bearing is the reverse of the direct line from the final position to port, found using the two legs as vectors — direct calculation gives approximately 246°(this question requires combining bearings with vector/trigonometry methods)
Q1383°(155° − 072°)
Q14Draw a circle of radius 5cm centred on A, and construct the perpendicular bisector of BC; P is where they intersect — generally 2 points (or 1 if the bisector is tangent to the circle, or 0 if it doesn't reach)
Q1590°(using alternate angles with the parallel North lines and angle sum in triangle ABC)
Common mistakes
Common Mistake 1
Rubbing out construction arcs
Exam markers need to see your compass arcs to award method marks — never erase them, even if the drawing looks messy.
Common Mistake 2
Writing bearings with fewer than three figures
A bearing of 60° must be written as 060°, not 60° — always use three figures, padding with a leading zero if needed.
Common Mistake 3
Getting the back bearing rule the wrong way round
Add 180° if the original bearing is less than 180°; subtract 180° if it's 180° or more. Adding when you should subtract (or vice versa) gives an answer outside the 0°–360° range.
Common Mistake 4
Confusing "equidistant from a line" with "equidistant from a point"
A fixed distance from a single point gives a circle. A fixed distance from a line segment gives a racetrack shape (straight sides plus rounded ends) — mixing these up is a common error.
Exam tips
💡 Exam Tip 1
Always use a sharp pencil and keep your compass width fixed
Accuracy matters for construction marks — a wobbly compass width between arcs is one of the most common reasons for losing marks.
💡 Exam Tip 2
Draw a North line at every point you're measuring a bearing from
Bearings are always measured from North at the specific point in question — a bearing "from A" needs North drawn at A, not anywhere else.
💡 Exam Tip 3
For combined loci, identify each condition separately first
Draw each locus on its own (a circle, a bisector, a racetrack) before combining them — trying to sketch the final shaded region in one step often leads to mistakes.
💡 Exam Tip 4
Use alternate angles for bearing problems with parallel North lines
Since all North lines point the same direction, they're parallel — alternate and co-interior angle facts frequently unlock bearing problems that look complicated at first.
Want to improve your grade faster?
If constructions, loci or bearings are still causing problems, Alamin's diagnostic approach identifies exactly which skills are missing and builds a targeted plan to address them — with AI-powered practice between sessions.