Constructions, Loci, Bearings & Scale Drawings

Module 4 · Lesson 4.04 · Spec G2, G15

Ruler-and-compass constructions, loci of points, three-figure bearings, and scale drawings — where your arcs are the evidence of your method.

What Makes This Lesson Different

Every other lesson so far has been about calculation. This one is about accurate drawing. You will be given a ruler and a pair of compasses — and you must use both.

What you need

Ruler marked in mm, pair of compasses, sharp pencil, and a protractor only for bearings. No protractor is allowed for the four standard constructions.

What you must NEVER do

Erase your construction arcs. They are the evidence of your method. AQA mark schemes explicitly require visible arcs — rubbing them out loses the method mark.

AQA Evidence

Nov 2024 Paper 1 Q11 on construction: "students struggled with this question; approximately one quarter achieved full marks; over 10% made no attempt." This is a topic where showing the working matters as much as the answer.

The Four Standard Constructions

Every GCSE paper tests these four. Learn the hand movements, not just the rules — your pencil traces the same shapes every time.

The four

1

Perpendicular bisector of a line segment AB — divides AB into two equal parts at 90°.

2

Angle bisector — splits an angle into two equal halves.

3

Perpendicular from a point to a line — drops a right angle from a point onto a line.

4

Triangle SSS (three sides given) — constructs a triangle from three given side lengths.

The Golden Rule

Compass radius stays the same between related arc steps. Change it and your construction breaks. If a step says "same radius", that means literally do not touch the compass.

Construction 1: Perpendicular Bisector

Divide line AB into two equal halves with a perpendicular through the midpoint.

Step-by-Step

Open compass to more than half of AB (otherwise arcs won't meet).

Place compass point on A. Draw an arc above AND below the line.

Without changing radius, place compass point on B. Draw arcs that cross the first ones.

Join the two intersection points with a ruler.

Why It Works

Both arc intersections are equidistant from A and B. The line joining them is therefore the locus of points equidistant from A and B — which is the perpendicular bisector.

Construction 2: Angle Bisector

Split an angle into two equal halves.

Step-by-Step

Place compass point on vertex V. Draw an arc that crosses both arms of the angle.

From each crossing point, draw an arc inside the angle — use the same radius for both.

Draw a straight line from V through the point where those two arcs meet.

Why It Works

The meeting point of the two small arcs is equidistant from both arms. The locus of points equidistant from two intersecting lines IS the angle bisector.

Construction 3: Perpendicular from a Point

Drop a perpendicular from a point P onto a line.

Step-by-Step

From P, draw an arc that crosses the line at two points Q₁ and Q₂.

From Q₁ and Q₂ with the same radius, draw arcs that meet on the opposite side of the line from P.

Join P to the meeting point. This line is perpendicular to the original line.

Tip

This is really just the perpendicular bisector of Q₁Q₂. The method is identical.

Variant — perpendicular AT a point ON the line

If the point P is on the line (not above or below it), the method adapts: set compass to any radius, mark two points either side of P on the line at that radius, then construct the perpendicular bisector of those two points. The bisector passes through P at 90°.

Construction 4: Triangle from Three Sides (SSS)

Given three side lengths, construct the unique triangle.

Step-by-Step (AB = 4 cm, AC = 5 cm, BC = 6 cm)

Draw the base AB to the given length (4 cm) with a ruler.

Set compass to the length of AC (5 cm). Place point on A. Draw an arc above AB.

Set compass to the length of BC (6 cm). Place point on B. Draw an arc that crosses the first.

The intersection is vertex C. Join A–C and B–C with a ruler.

Check the Triangle Inequality

Each side must be less than the sum of the other two (4 < 5 + 6 ✓, 5 < 4 + 6 ✓, 6 < 4 + 5 ✓). If this fails, the arcs never meet and no triangle exists.

Bonus — Constructing a 60° angle

An equilateral triangle is just SSS with all three sides equal. Use it to construct a 60° angle: draw base AB, set compass to AB's length, arc from A and arc from B, join A (or B) to the intersection. Every angle in the resulting triangle is 60° — useful whenever an exam asks you to construct 60° without a protractor.

Loci: The Four Key Cases

A locus (plural: loci) is the set of all points satisfying a given condition. Every GCSE locus question reduces to one of these four.

ConditionLocusConstruction

Equidistant from a fixed point PA circle centred at PCompass set to the given distance, centred at P

A fixed distance from a lineTwo parallel lines (one each side), plus semicircles at the ends if the line is a segmentRuler, then compass for the end caps

Equidistant from two points A, BThe perpendicular bisector of ABConstruction 1

Equidistant from two intersecting linesThe angle bisector of the angle between themConstruction 2

Regions (shading)

Questions often ask for a region, not a line — e.g. "more than 10 m from P" is outside the circle of radius 10 m around P; "closer to A than to B" is on A's side of the perpendicular bisector. Shade the region; don't just draw the boundary.

Worked Example — Locus Region01

A farmer's rectangular field has corners at P, Q, R, S. A goat is tethered at P. The rope is 10 m long. Shade the region the goat can reach. Then shade the region that is more than 5 m from the fence QR. [3 marks]

M

Method: (a) Locus of points ≤ 10 m from P is a quarter-circle of radius 10 m centred at P (quarter because P is a corner — the rest is outside the field). (b) Locus of points more than 5 m from fence QR is the region to the left of a line parallel to QR at distance 5 m.

C

Calculate / Construct: Using the scale 1 cm = 1 m: draw a 10 cm arc from P inside the field. Draw a line parallel to QR, 5 cm to the left of QR. Shade the two required regions (or the intersection if asked).

E

Evaluate / Answer: Both loci are shown. Leave the compass arcs and the construction line visible — that's where the method mark lives.

Mark split: B1 for the quarter-circle locus, B1 for the parallel-line locus, B1 for correct shading of both regions.

Bearings: The Rules

A bearing is a direction, written as an angle. It follows two rigid rules.

Rule 1 — Measure clockwise from North

Start from the North line pointing UP. Rotate clockwise. The angle you turn through is the bearing.

Rule 2 — Always write three digits

Pad with zeros at the front if the value is under 100°. 070°, not 70°. 045°, not 45°. 120° is already three digits — no padding needed. 005° for very small bearings.

Compass Points → Bearings

North = 000° (or 360°), East = 090°, South = 180°, West = 270°. NE = 045°, SE = 135°, SW = 225°, NW = 315°.

Measuring and Drawing Bearings

Every bearing question needs a North line drawn at the point you're measuring from.

To measure a bearing

Draw a North line (straight up) at the starting point.

Place the centre of your protractor on the starting point with 0° along North.

Read the angle clockwise from North to the target.

Write the answer as three digits with a degree sign.

To draw a bearing

At the starting point, draw a North line.

Use a protractor to measure the bearing clockwise from North.

Draw a line (often with a ruler and to a given scale distance) at that angle.

Common Errors

Measuring from East (instead of North): makes every bearing 90° wrong. Measuring anti-clockwise: makes every bearing 360°−actual wrong. Forgetting the three-digit format: loses the format mark.

Back Bearings

If the bearing from A to B is given, the back bearing (bearing from B to A) differs by exactly 180°.

Back bearing = forward bearing ± 180°

Rule of Thumb

If the forward bearing is less than 180°: add 180°. (e.g. 070° → 250°)

If the forward bearing is 180° or more: subtract 180°. (e.g. 245° → 065°)

Why It Works

The two North lines at A and B are parallel. The angle the line AB makes with North at A plus the angle AB makes with North at B is always 180° (co-interior angles on parallel lines — from 4.02).

Worked Example — Back Bearing02

A ship sails from port A on a bearing of 070° to port B, a distance of 80 km. Work out the bearing from B back to A. [1 mark]

M

Method: Forward bearing 070° is less than 180°, so add 180° to get the back bearing.

C

Calculate: Back bearing = 70 + 180 = 250°.

E

Evaluate / Answer: bearing from B to A = 250°.

Mark split: B1 for 250° (must be in three-digit form — "250°" not "250" and not "2.5 hundred").

Scale Drawings

A scale drawing represents a real-world object accurately, but at a different size. The scale tells you how to convert between the drawing and reality.

Reading a scale

"1 cm represents 5 m" (ratio form): every 1 cm on the drawing = 5 m in real life. A line 6 cm long represents 30 m.

"1 : 500" (scale factor): every 1 unit on the drawing = 500 of the same unit in real life. 1 cm on drawing = 500 cm = 5 m.

Workflow

Convert the given real-world distance into the drawing's unit using the scale.

Draw the line to that length.

When reading from the drawing, measure in cm and multiply by the scale to get the real distance.

Units Matter

Scales like 1:500 have no units — same unit on both sides. Scales like "1 cm = 5 m" mix units; convert carefully. Always keep a cm ruler handy for these questions.

Worked Example — Bearings + Scale Drawing03

A ship sails from port A on a bearing of 060° for 40 km, then turns and sails on a bearing of 150° for 30 km to reach port C. Using a scale of 1 cm = 5 km, construct the journey. Then measure the direct distance from A to C. [4 marks]

M

Method: Convert each distance to cm: 40 km = 8 cm, 30 km = 6 cm. Draw AB on bearing 060° (measured from North at A) of length 8 cm. At B, draw a new North line and measure bearing 150° clockwise from it. Draw BC of length 6 cm. Measure AC with a ruler and convert back to km.

C

Construct: Mark A. Draw North line at A. Use protractor for 60° clockwise. Ruler: 8 cm to B. Mark B. Draw a new North line at B (parallel to the one at A). Protractor for 150° clockwise from the new North line. Ruler: 6 cm to C. Measure AC.

E

Evaluate: If AC measures 10 cm on the drawing, then AC in real life = 10 × 5 = 50 km. (Exact value depends on accurate construction — accept 48–52 km.)

Mark split: B1 bearing 060° drawn correctly, B1 length AB 8 cm, B1 bearing 150° from B, B1 final AC measured and converted. Show all North lines — examiners look for them.

Examiner Alerts

Alert 1 — Erasing construction arcs

The most common construction mistake. AQA mark schemes state "appropriate arcs visible — B1". No arcs = no method mark, even if the answer line is perfect.

Alert 2 — Two-digit bearings

Writing "70°" instead of "070°" is a format error. AQA consistently deducts a mark. Always pad bearings to three digits.

Alert 3 — Missing North line

Every bearing diagram needs a North line at each point where a bearing is measured. Forgetting this at a second or third point is a classic multi-step error.

Alert 4 — Drawing the boundary of a locus, not the region

"Shade the region where …" needs the region shaded (not just the boundary line drawn). Read the command word: "shade" ≠ "draw".

Alert 5 — Using a protractor where a compass is required

The four standard constructions must use compass only. Measuring angles with a protractor — even if accurate — scores zero for method.

Summary

Four standard constructions

Perpendicular bisector — arcs from A and B with radius > ½AB.

Angle bisector — arc from vertex, then arcs from crossings.

Perpendicular from a point — arc from P, then perpendicular bisector of the crossings.

Triangle SSS — base first, then arcs from each endpoint.

Four loci

From a point → circle. From a line → parallel lines. From two points → perpendicular bisector. From two lines → angle bisector.

Bearings

Clockwise from North, three digits.

Back bearing = forward bearing ± 180°.

Every diagram needs a North line at each measuring point.

Scale drawings

1 cm represents k units → every drawn cm is k units in reality.

Multiply to go drawing → real , divide to go real → drawing .

Never Forget

Keep your arcs.Three digits.North line first. These three habits save most of the marks in this topic.

Checkpoint

You should now be able to

Perform all four standard constructions with ruler and compass, leaving arcs visible.

Describe and draw the four standard loci.

Shade regions based on locus conditions ("more than X from P", "closer to A than B").

Read, write, measure and draw bearings in three-digit form, clockwise from North.

Convert between forward and back bearings using ± 180°.

Interpret scale drawings ("1 cm = 5 km" or "1 : 500") in both directions.

Test yourself in under two minutes

1

Ship A sails on bearing 215° from port. What's the back bearing?

2

Shade the set of points that are both within 3 cm of P AND closer to A than to B.

3

On a 1 : 200 plan, a room measures 4 cm by 6 cm. What are its real dimensions in metres?

4

Which two standard constructions give you a locus equidistant from two points? And from two intersecting lines?

Answers: 1) 215 − 180 = 035°. 2) Circle of radius 3 cm centred at P, intersected with the half-plane on A's side of the perpendicular bisector of AB. 3) 4 cm × 200 = 800 cm = 8 m; 6 cm × 200 = 12 m, so 8 m × 12 m. 4) Perpendicular bisector (two points); angle bisector (two lines).

Next Lesson (4.05)

We move to transformations: rotation, reflection and translation — and how to describe each one fully so the examiner awards every mark.