Bearings, Loci and Constructions

What you will learn

• How to measure and draw three-figure bearings clockwise from north.
• How to use a scale diagram to solve route and distance questions.
• How to find reverse bearings by adding or subtracting 180°.
• How to recognise common loci: fixed distance, equal distance and regions.
• How to construct perpendicular bisectors and angle bisectors.
• How to leave clear construction marks and label your final answer.

Why this matters

These skills turn written instructions into precise diagrams. They appear in map routes, safety zones, mobile-signal regions, treasure-map problems, navigation and construction-style exam questions.

Prior knowledge

You should already be comfortable with:

• using a ruler, protractor and pair of compasses,
• measuring angles from a straight north line,
• using scale drawings and map scales,
• recognising perpendicular and parallel lines,
• understanding a circle as all points the same distance from its centre.

Clear explanation

Bearings are measured clockwise from north

A bearing is always written as three digits. Draw a north line at the starting point, measure clockwise, then draw the route line. For example, a bearing of 065° means 65° clockwise from north.

Reverse bearings are 180° different because the direction has turned to face back along the same line. Add 180° if the bearing is less than 180°; subtract 180° if it is greater than 180°.

Loci are sets of possible positions

A locus is a path or region that follows a rule. Points a fixed distance from one point form a circle. Points a fixed distance from a straight line form two parallel lines. Points equal distance from two points lie on the perpendicular bisector. Points equal distance from two lines lie on the angle bisector.

Region questions usually add words such as less than, more than, within, outside, nearer to or equal distance from. Draw the boundary first, decide whether the boundary line should be included, then shade only the part that satisfies every rule in the question.

Angle bisectors are easy to miss because they are not always drawn between two named points. If a question says equal distance from two intersecting lines, construct the angle bisector of the angle or angles that fit the region. If the question says nearer to one point than another, the perpendicular bisector is the boundary and the correct side of it is the answer.

Constructions need visible arcs

Use compasses without changing the radius when the construction needs equal distances. Keep the arcs on the page: they show how the bisector or perpendicular line was made. On a scale diagram, label the scale length you used before measuring the final real distance.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For bearings, loci and constructions, marks usually come from drawing a north line at the correct point, measuring clockwise, handling reverse bearings as 180° changes, using the stated scale, naming the correct locus rule, deciding which side or region satisfies the wording, keeping compass arcs visible, and labelling the final route, line, circle or shaded region clearly.

Common mistakes

• Measuring from the wrong place: the north line belongs at the starting point of the bearing.
• Measuring anticlockwise: GCSE bearings are measured clockwise from north.
• Writing two-digit bearings: use 038°, not 38°.
• Rubbing out construction arcs: arcs are evidence of the compass method.
• Mixing up loci: fixed distance from a point is a circle; equal distance from two points is a perpendicular bisector.
• Hiding the working diagram: leave north lines, construction arcs, scale labels and boundary lines visible unless the question tells you otherwise.

Extension challenge

On a map, town B is 10 km from town A on a bearing of 060°. Town C is 6 km from A and is equal distance from A and B. Describe how to construct the possible positions of C.

Exam-board guidance

Bearings, loci and constructions are core GCSE Maths geometry skills across the supported boards. They may appear as direct drawing tasks or as map, route, region and scale-diagram problems.

Next lesson

Next, move from accurate drawing to moving shapes in Transformations.