Bearings, Scale Drawing & ConstructionsEdexcel IGCSE Maths: Revision notes

Section 1

What is a three-figure bearing?

A bearing is a direction given as an angle measured clockwise from North, always written using three figures (e.g. $007^\circ$ , $090^\circ$ , $315^\circ$ ).

Rules:

Always measure clockwise from North.
Always write three digits, adding leading zeros if needed (e.g. $05^\circ$ becomes $005^\circ$ ).
Bearings range from $000^\circ$ to $360^\circ$ (never negative, never above $360^\circ$ ).

To find a bearing from a diagram: draw a North line at the starting point, then measure the clockwise angle to the object.

Key termsbearingNorth lineclockwise

Common mistake

Writing a bearing as $47^\circ$ instead of $047^\circ$ loses marks — always pad to three digits.

Example

A ship is due East of a lighthouse. Its bearing from the lighthouse is $090^\circ$ . A ship due South has bearing $180^\circ$ .

Section 2

How do I find a back bearing?

A back bearing is the bearing of the return journey — from B back to A, if you know the bearing from A to B.

Use these rules:

If the bearing A to B is less than $180^\circ$ , add $180^\circ$ to get the back bearing.
If the bearing A to B is $180^\circ$ or more, subtract $180^\circ$ .

This works because North lines at A and B are parallel, so alternate/co-interior angle facts link the two bearings — the back bearing is always exactly $180^\circ$ different from the original.

Key termsback bearingalternate angles

Exam tip

Quick check: if the original bearing is small (under $180^\circ$ ), the back bearing must be over $180^\circ$ , and vice versa.

Example

The bearing of B from A is $065^\circ$ . The bearing of A from B is $065^\circ + 180^\circ = 245^\circ$ .

Section 3

How do scale drawings and maps work?

A scale drawing represents real distances at a fixed ratio, e.g. $1 : 50000$ or " $1\text{ cm represents } 2\text{ km}$ ".

To convert:

Map distance to real distance: multiply the measured length by the scale factor.
Real distance to map distance: divide the real length by the scale factor.

Always convert units carefully — scales are often given as $1\text{ cm} : n\text{ km}$ , so convert km to cm ( $1\text{ km} = 100000\text{ cm}$ ) before working with ratio scales like $1 : 50000$ .

Key termsscalescale factorratio

Common mistake

Forgetting to convert units before applying the scale — mixing cm and km gives answers that are wildly wrong.

Example

Scale $1\text{ cm} : 5\text{ km}$ . A map distance of $3.4\text{ cm}$ represents $3.4 \times 5 = 17\text{ km}$ in real life.

Section 4

How do I do bearings and scale drawing questions together?

Many exam questions ask you to draw a scale diagram using a given bearing and scale, then measure a distance or bearing from it.

Method:

Choose your scale (e.g. $1\text{ cm} = 10\text{ km}$ ) and mark the starting point.
Draw a North line (vertical arrow) at that point.
Use a protractor to measure the given bearing clockwise from North.
Draw the line to the correct scaled length using a ruler.
Repeat from the new point if there is a second leg of the journey.
To answer the question, measure the required length or angle from your accurate diagram.

Key termsprotractorscale diagram

Exam tip

Draw a fresh North line at every point where a new bearing is given — do not assume North lines are already there.

Think of it like this

Think of it like giving satnav directions: 'go this exact compass direction for this exact distance, then repeat' — the scale diagram is just that journey drawn accurately on paper.

Section 5

What compass constructions do I need to know?

Constructions must be drawn using only a ruler and compasses (no protractor), leaving all construction arcs visible as proof of method.

Key constructions:

Perpendicular bisector of a line: open compasses to more than half the line's length, draw arcs from both ends above and below the line, join the two arc intersection points.
Angle bisector: from the vertex, draw an arc crossing both arms; from each crossing point draw arcs that intersect; join the vertex to that intersection.
Perpendicular from a point to a line / at a point on a line: use arcs centred on the point to mark equal distances on the line, then bisect between those marks.
Loci: a locus is a set of points satisfying a rule (e.g. "equidistant from A and B" = perpendicular bisector of AB; "equidistant from two lines" = angle bisector; "exactly $x\text{ cm}$ from a point" = a circle of radius $x$ ).

Key termsperpendicular bisectorangle bisectorlocusconstruction arcs

Common mistake

Rubbing out construction arcs loses method marks — examiners need to see the compass arcs to award marks, even if the final line looks right.

Exam tip

For a region "closer to A than to B", shade the correct side of the perpendicular bisector of AB rather than just drawing the line.

Must Know

Bearings are always measured clockwise from North and written as three figures (e.g. $008^\circ$ ).
Back bearing rule: add $180^\circ$ if the original bearing is under $180^\circ$ ; subtract $180^\circ$ if it is $180^\circ$ or over.
Scale drawings: multiply map distance by the scale factor for real distance; divide real distance by the scale factor for map distance — convert units first.
Draw a new North line at every point where a bearing is given, using a protractor for the angle and a ruler for the scaled length.
Constructions use only ruler and compasses — always leave the arcs visible.
Perpendicular bisector = equidistant from two points; angle bisector = equidistant from two lines; fixed distance from a point = a circle.

That's the notes covered.

Carry on to the next subtopic.