Bearings | FractionRush

Welcome to Bearings!

Bearings are the navigator's language — used in aviation, marine navigation, orienteering and surveying. They give a precise, unambiguous way to describe direction using a single three-digit angle. This topic is a regular IGCSE exam question, often combined with trigonometry for high-mark problems.

Three-figure bearing: measured clockwise from North | Back bearing: ± 180° | Always write 3 digits: 045°, not 45°

Learning Objectives

State and use the definition of a three-figure bearing
Convert between compass directions (N, NE, S, SW…) and bearings
Read and measure bearings from scale diagrams
Calculate reverse (back) bearings
Use alternate angles and parallel lines to find unknown bearings
Solve multi-step bearing problems
Combine bearings with right-angled trigonometry
Apply the sine rule and cosine rule to bearing problems

Three-Figure Bearings

Clockwise from North, 000°–359°

Compass Directions

N=000°, E=090°, S=180°, W=270°

Back Bearings

Reverse direction: ± 180°

Alternate Angles

Parallel North lines → co-interior, alternate

Right-Angled Trig

SOHCAHTOA with bearing diagrams

Sine & Cosine Rule

Non-right-angled bearing problems

Learn 1 — Three-Figure Bearings

What is a Bearing?

A bearing is an angle used to describe a direction. Bearings follow three strict rules that you must always apply:

The Three Rules of Bearings:
1. Always measured from North
2. Always measured clockwise
3. Always written as 3 digits (e.g. 045°, not 45°; 007°, not 7°)

Memory trick: NCC — North, Clockwise, Characters (3 digits). If your angle has fewer than 3 digits, pad with leading zeros: 9° → 009°, 72° → 072°.

Compass Directions and Their Bearings

The main compass points each correspond to an exact bearing. Learn these — they are used as reference points in all bearing problems.

Compass Direction	Bearing	Compass Direction	Bearing
North (N)	000°	South (S)	180°
North-East (NE)	045°	South-West (SW)	225°
East (E)	090°	West (W)	270°
South-East (SE)	135°	North-West (NW)	315°
North-North-East (NNE)	022.5°	South-South-West (SSW)	202.5°
East-North-East (ENE)	067.5°	West-South-West (WSW)	247.5°

Visualise a clock face overlaid on a compass. 12 o'clock = North = 000°. 3 o'clock = East = 090°. 6 o'clock = South = 180°. 9 o'clock = West = 270°.

Reading Bearings from Diagrams

When a bearing diagram shows a North arrow and an angle, read the bearing by measuring clockwise from the North arrow to the direction line.

Example: Reading a bearing
A North arrow points upward. The direction to point B is drawn 65° clockwise from North.
The bearing of B from the starting point = 065°

Example: Writing a bearing as a compass description
A bearing of 310° — where does this point?
310° is between 270° (West) and 360° (North), so it points North-West.
More precisely: 360° − 310° = 50° west of North → described as "N50°W" in older notation.

Bearings are always measured from the North line at the starting point, not from the destination point. "The bearing of B from A" means: stand at A, face North, rotate clockwise to face B — that angle is the bearing.

Drawing Bearings on Diagrams

To draw the bearing of point B from point A at bearing θ°:

Steps to draw a bearing:
1. Mark point A and draw a vertical North line upward from A
2. Place your protractor at A with 0° on the North line
3. Measure θ° clockwise from North
4. Draw a ray from A in that direction
5. Mark point B on the ray at the required distance (using the scale)

Example: Draw the bearing 130° from point A.
North is up (0°). 90° takes us to East. We continue 40° more past East (towards Southeast).
The line from A points into the South-East quadrant, 40° past due East.

When a bearing is greater than 180°, your protractor only goes to 180°. Measure 360° − θ° anticlockwise from North instead, OR measure (θ° − 180°) from South going clockwise.

Learn 2 — Back Bearings & Multi-Step Problems

Reverse (Back) Bearings

If A is on a bearing of θ from B, then B is on the back bearing (reverse bearing) from A. The two North lines at A and B are parallel (both pointing North), so you can use alternate angles.

If bearing < 180°: Back bearing = bearing + 180°
If bearing ≥ 180°: Back bearing = bearing − 180°

Example 1: The bearing of B from A is 065°. Find the bearing of A from B.
065° < 180°, so back bearing = 065° + 180° = 245°

Example 2: The bearing of B from A is 230°. Find the bearing of A from B.
230° ≥ 180°, so back bearing = 230° − 180° = 050°

Why does this work? The North lines at A and B are parallel. The line AB acts as a transversal. The bearing and its back bearing are co-interior angles (same-side interior angles), which add to 360° when both measured clockwise — which simplifies to the ±180° rule.

Parallel North Lines and Angle Relationships

All North lines are parallel to each other. This means you can use the full toolkit of parallel-line angle properties when solving bearing problems.

Alternate angles (Z-angles): When a line cuts two parallel North lines, alternate angles are equal.

Co-interior angles (C-angles): Co-interior angles between parallel lines add to 180°.

Corresponding angles (F-angles): Corresponding angles between parallel lines are equal.

Example: A is due North of C. B is on a bearing of 070° from A. Find the bearing of B from C.
Draw North lines at A and C (parallel). The bearing 070° is measured at A.
The angle at C (from North to CB) = 070° (alternate angles, since A is due North of C and the North lines are parallel).
Bearing of B from C = 070°

The alternate angle trick only works directly when A is directly North/South of C (so the two North lines are collinear with the vertical). In more general cases, draw a careful diagram and identify which angle relationship applies.

Multi-Step Bearing Problems

These questions involve two or more legs of travel. The key is to draw a clear diagram, mark all North lines, and work out each angle step by step.

Example: A ship sails from P on a bearing of 040° to Q, then on a bearing of 160° to R. Find the bearing of R from P.
Step 1 — Draw P with North line. Draw PQ at 040°.
Step 2 — Draw North line at Q (parallel to North at P). Draw QR at 160° from Q's North.
Step 3 — The bearing of R from P requires finding angle NPR (N = North at P).
The bearing of R from P depends on the geometry — use alternate angles.
Angle at Q between PQ and North = 040° (alternate angles, parallel North lines).
Angle PQR = 160° − 040° = 120° (interior angle of triangle at Q).
Use further geometry or trigonometry to complete the calculation.

For multi-step bearing problems: (1) Always draw and label a diagram. (2) Mark all North lines. (3) Write down the bearing at each point. (4) Use parallel-line angles to find interior angles. (5) Use trig if distances are involved.

Finding Angles in a Triangle from Bearings

Standard method to find the interior angle at a point:
If a ship travels on bearing α to reach point B, then turns to bearing β:
— The exterior angle at B (between the back-bearing of the first leg and the next leg) = β
— The interior angle at B = (back bearing of BA) − β, or 360° − (back bearing of BA) + β, depending on the diagram.

Always check your diagram rather than applying a formula blindly.

Learn 3 — Bearings with Trigonometry

Right-Angled Trigonometry with Bearings

When a bearing problem produces a right angle (e.g., someone travels East then North), use SOHCAHTOA. The bearing tells you the direction of the hypotenuse.

Example: A walks 5 km due East, then 12 km due North. Find the bearing of A's final position from the start.
Draw a right-angled triangle: horizontal leg = 5 km (East), vertical leg = 12 km (North).
The angle from North to the hypotenuse (bearing direction):
tan(θ) = opposite/adjacent = 5/12 (East component over North component)
θ = arctan(5/12) = 22.6°
Bearing = 023° (to 3 sig. figs., measured clockwise from North)

Key insight: In a bearing right-triangle, the North component is always the side adjacent to the bearing angle (since bearing is measured from North). The East/West component is the opposite side. So: tan(bearing) = East component ÷ North component.

Example: A plane flies on bearing 035° for 200 km. How far North and how far East has it travelled?
North component = 200 cos(35°) = 200 × 0.8192 = 163.8 km
East component = 200 sin(35°) = 200 × 0.5736 = 114.7 km

General rule for a leg of bearing θ and distance d:
Northward displacement = d cos(θ)
Eastward displacement = d sin(θ)
(Negative values mean South or West respectively.)

The Sine Rule with Bearings

When a bearing problem produces a non-right-angled triangle, use the sine rule or cosine rule. First, find the interior angles of the triangle using the bearing information and parallel-line properties.

Sine Rule: a/sin A = b/sin B = c/sin C

Example: From lighthouse L, ship A is on bearing 040° and ship B is on bearing 115°. LA = 8 km. Find AB if angle LAB = 72°.
Angle at L between LA and LB = 115° − 040° = 75°
Angle at A = 72° (given)
Angle at B = 180° − 75° − 72° = 33°
By sine rule: AB/sin(75°) = 8/sin(33°)
AB = 8 × sin(75°)/sin(33°) = 8 × 0.9659/0.5446 = 14.2 km

The Cosine Rule with Bearings

Cosine Rule: c² = a² + b² − 2ab cos C | cos C = (a² + b² − c²) / (2ab)

Example: Port P, town A is 15 km on bearing 050°, town B is 20 km on bearing 115°. Find distance AB.
Angle APB = 115° − 050° = 65° (angle at P between the two bearings)
By cosine rule: AB² = 15² + 20² − 2(15)(20)cos(65°)
AB² = 225 + 400 − 600 × 0.4226
AB² = 625 − 253.6 = 371.4
AB = √371.4 = 19.3 km

The angle used in the cosine rule is the angle between the two sides you know — which is the angle at the starting point, found by subtracting the two bearings (or adding/subtracting using the diagram).

Scale Drawings to Find Distances

For problems that don't require exact answers, a scale drawing can find distances and bearings graphically.

Steps for a scale drawing:
1. Choose a suitable scale (e.g. 1 cm : 5 km)
2. Draw the North arrow at the starting point
3. Use a protractor to draw the bearing lines
4. Measure lengths using the scale
5. Measure the bearing of the final direction using a protractor

Scale drawings give approximate answers only. Use trigonometry for exact answers in exam questions that ask you to "calculate".

Example 1 — Reading and Writing a Three-Figure Bearing

Q: The bearing of town B from town A is shown on a diagram with angle 48° measured clockwise from North. Write the bearing of B from A.

M1: The angle is 48°. Since this is less than 100, we need a leading zero to make it 3 digits.

A1: Bearing of B from A = 048°

Note: Never write 48° as a bearing — the 3-digit rule is compulsory in IGCSE.

Example 2 — Back Bearing

Q: The bearing of Q from P is 127°. Find the bearing of P from Q.

M1: Since 127° < 180°, add 180°: back bearing = 127° + 180° = 307°

A1: Bearing of P from Q = 307°

Check: 307° is in the North-West quadrant. Q is to the south-east of P (bearing 127°), so looking back from Q, P should be in the north-west direction. ✓

Example 3 — Multi-Step Problem Using Alternate Angles

Q: From port A, a ship sails on bearing 052° to reach buoy B, which is due East of lighthouse C. The bearing of B from C is 270°... wait — Find the bearing of C from B, given the bearing of B from A is 052° and C is directly south of A.

Setup: A is north of C (C is due South of A). Bearing of B from A is 052°.

M1: Draw North lines at A and C (parallel). The angle between North at A and AB = 052°.

M1: Since A is directly north of C, the line AC is a transversal crossing two parallel North lines. The alternate angle at C = 052° (alternate interior angles).

A1: Bearing of B from C = 052°

Example 4 — Right-Angled Trig with Bearings

Q: A boat travels 8 km due North, then 6 km due East. Find the bearing of the final position from the start, and the total distance from the start.

M1: Draw: vertical leg = 8 km (North), horizontal leg = 6 km (East). This is a right-angled triangle.

M1: tan(θ) = East/North = 6/8 = 0.75 → θ = arctan(0.75) = 36.87°

A1: Bearing = 037° (to nearest degree) A1

M1: Distance = √(8² + 6²) = √(64 + 36) = √100 = 10 km A1

Example 5 — Cosine Rule with Bearings

Q: From harbour H, ship A is 12 km on bearing 060°. Ship B is 9 km on bearing 135°. Calculate the distance AB.

M1: Angle AHB = 135° − 060° = 75°

M1: By cosine rule: AB² = 12² + 9² − 2(12)(9)cos(75°)

AB² = 144 + 81 − 216 × cos(75°) = 225 − 216 × 0.2588 = 225 − 55.9 = 169.1

A1: AB = √169.1 = 13.0 km A1

Example 6 — Sine Rule to Find a Bearing

Q: Town A is 10 km from town P on bearing 040°. Town B is north of A with AB = 7 km. Find the bearing of B from P.

M1: Sketch the triangle PAB. Angle at P between North and PA = 040°. AB = 7 km, PA = 10 km, angle PAB = 180° − 040° = 140° (back bearing direction interior angle).

M1: By sine rule: sin(APB)/AB = sin(PAB)/PB — first find PB using cosine rule or identify the geometry.

M1: Using the correct triangle, angle PBA found via sine rule, then bearing of B from P = 040° − angle APB.

A1: Full method: sin(BPA)/7 = sin(140°)/PB (need PB first from cosine rule), then bearing = 040° ± angle. Follow the diagram carefully. M3 A1

Common Mistakes in Bearings

These are the errors examiners see most often. Understanding why they are wrong helps you avoid them.

Mistake 1 — Measuring Anticlockwise Instead of Clockwise

✗ Wrong: The angle to B from North is 70° to the left (anticlockwise), so bearing = 070°

✓ Correct: Bearings are ALWAYS measured clockwise from North. If the angle is to the left (west side), the bearing is 360° − 70° = 290°

This is the most common mistake. Always rotate clockwise from North when measuring or drawing a bearing. A bearing to the West is between 180° and 360°, never a small number.

Mistake 2 — Not Writing 3 Digits

✗ Wrong: The bearing of B from A is 45°

✓ Correct: The bearing of B from A is 045°

IGCSE mark schemes specifically require three digits. Writing 45° instead of 045° can lose the accuracy mark. Always add leading zeros: 9° → 009°, 72° → 072°. Angles from 100° to 359° already have 3 digits.

Mistake 3 — Confusing "Bearing of A from B" with "Bearing of B from A"

✗ Wrong: "The bearing of B from A is 065°, so the bearing of B from A is also 065°" (copying when asked the reverse)

✓ Correct: "Bearing of B from A is 065°" means stand at A and look towards B. "Bearing of A from B" means stand at B and look back towards A → back bearing = 065° + 180° = 245°

Carefully read which point you are standing at ("from"). The word "from" tells you where the North arrow goes. The word preceding "from" is the direction you face.

Mistake 4 — Using the Wrong Angle in the Sine/Cosine Rule

✗ Wrong: Two bearings are 050° and 130° from the same point. Using angle = 130° in the cosine rule.

✓ Correct: The angle between the two directions = 130° − 050° = 80°. This is the interior angle of the triangle at the starting point.

Always subtract the two bearings (larger minus smaller) to find the angle between the two lines at the common point. Draw a diagram to confirm which angle is which.

Mistake 5 — Back Bearing Rule Applied Incorrectly

✗ Wrong: Bearing is 250°, so back bearing = 250° + 180° = 430°

✓ Correct: 250° ≥ 180°, so back bearing = 250° − 180° = 070°

The back bearing must always be between 000° and 359°. If adding 180° takes you above 360°, subtract 180° instead. Always check: a bearing and its back bearing must differ by exactly 180°.

Mistake 6 — Forgetting to Draw a North Line at Every Point

✗ Wrong: Only drawing one North arrow and measuring all bearings from it

✓ Correct: Draw a North line at every point from which a bearing is given or required. All North lines are parallel — this lets you use alternate and co-interior angles.

In multi-step bearing problems, each point has its own North line. The parallel North lines at different points are the key to finding interior triangle angles.

Key Formulas — Bearings

Formula / Rule	When to Use
Back bearing (bearing < 180°): Back bearing = bearing + 180°	Finding the reverse direction
Back bearing (bearing ≥ 180°): Back bearing = bearing − 180°	Finding the reverse direction
Angle between two bearings: = larger bearing − smaller bearing	Finding interior angle at a point
North component: d cos θ	Right-angled trig: how far North
East component: d sin θ	Right-angled trig: how far East
Bearing from components: θ = arctan(East ÷ North)	Finding bearing from N/E displacements
Sine Rule: a/sin A = b/sin B = c/sin C	Two sides & non-included angle, or two angles & one side
Cosine Rule (side): c² = a² + b² − 2ab cos C	Two sides & included angle — find third side
Cosine Rule (angle): cos C = (a² + b² − c²) / (2ab)	Three sides — find an angle

Key compass bearings: N=000° · NE=045° · E=090° · SE=135° · S=180° · SW=225° · W=270° · NW=315°

Three rules of bearings: From NORTH · Measured CLOCKWISE · Always 3 DIGITS

Useful Angle Relationships

Alternate angles (Z-angles): Equal, formed between parallel North lines and a transversal.
Co-interior angles (C-angles): Sum to 180°, between parallel North lines on the same side.
Corresponding angles (F-angles): Equal, both on the same side of a transversal.
Angles on a straight line: Sum to 180°.
Angles around a point: Sum to 360°.

SOHCAHTOA Reminder

sin θ = Opposite / Hypotenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent

In a bearing right-triangle: the bearing angle is at the start point, between North (adjacent) and the direction of travel (hypotenuse). So:
North component (adjacent) = d cos θ East component (opposite) = d sin θ

Interactive Bearing Compass

Enter a bearing to see it drawn from North on the compass. You can also add a second bearing to see the angle between them.

Bearing 1 (°): Distance (km):

Bearing 2 (°):

Enter a bearing (0–359) and click "Draw Bearing" to begin.

Exercise 1 — Reading, Writing & Compass Directions

1. Write the bearing of North-East as a three-figure bearing (°).

2. Write the bearing of South-West as a three-figure bearing (°).

3. What bearing corresponds to due South? (°)

4. What bearing corresponds to due West? (°)

5. A bearing is described as "72° clockwise from North". Write this as a three-figure bearing (°).

6. What bearing corresponds to North-West? (°)

7. A bearing is described as "8° clockwise from North". Write this as a three-figure bearing (°).

8. What bearing corresponds to South-East? (°)

Exercise 2 — Back (Reverse) Bearings

1. The bearing of B from A is 040°. Find the bearing of A from B (°).

2. The bearing of B from A is 110°. Find the bearing of A from B (°).

3. The bearing of Q from P is 165°. Find the bearing of P from Q (°).

4. The bearing of Q from P is 200°. Find the bearing of P from Q (°).

5. The bearing of B from A is 285°. Find the bearing of A from B (°).

6. The bearing of B from A is 330°. Find the bearing of A from B (°).

7. The bearing of C from D is 073°. Find the bearing of D from C (°).

8. The bearing of X from Y is 252°. Find the bearing of Y from X (°).

Exercise 3 — Angles Between Bearings & Parallel North Lines

1. Two paths leave from point P on bearings 030° and 090°. Find the angle between the two paths (°).

2. From point A, bearing to B is 055° and bearing to C is 120°. Find angle BAC (°).

3. From P, bearing to Q is 040°. C is due South of P. What is the bearing of Q from C (alternate angles apply)? (°)

4. Two bearings from the same point are 145° and 260°. Find the angle between them (°).

5. A ship sails on bearing 070°. It then turns to sail on bearing 160°. Through what angle did it turn, measured clockwise? (°)

6. From A, bearing to B is 035°. From B, bearing to C is 035°. A is due North of D, and D is at the same latitude as B. Find the bearing of B from D using alternate angles. (°)

7. Bearings from X to Y is 310° and from X to Z is 050°. Find the angle YXZ going clockwise from Z to Y. (°)

8. From harbour H, boat A is on bearing 025° and boat B is on bearing 205°. Are A, H, B collinear (on a straight line)? Enter 1 for Yes, 0 for No.

Exercise 4 — Right-Angled Trigonometry with Bearings

1. A plane flies 100 km due North then 100 km due East. Find the bearing of its final position from the start (° to nearest degree).

2. A walker goes 6 km due East then 8 km due North. Find the bearing of the final position from the start (° to nearest degree).

3. A boat sails on bearing 090° for 12 km. How far East of the start is it? (km)

4. A plane flies on bearing 030° for 200 km. How far North has it travelled? (km, 1 d.p.)

5. A ship sails on bearing 030° for 200 km. How far East has it travelled? (km, 1 d.p.)

6. A town is 5 km East and 12 km North of a city. Find the bearing of the town from the city (° to nearest degree).

7. A hiker walks 10 km North then 10 km East. What is the straight-line distance from the start? (km, 1 d.p.)

8. A plane flies on bearing 045° for 50 km. How far East of the start is it? (km, 1 d.p.)

Exercise 5 — Sine Rule & Cosine Rule with Bearings

1. From P, A is 10 km on bearing 040° and B is 15 km on bearing 100°. Find the angle APB (°).

2. From harbour H, ship A is 8 km on bearing 060° and ship B is 6 km on bearing 150°. Find the distance AB (km, 1 d.p.).

3. From P, A is 12 km on bearing 050° and B is 12 km on bearing 110°. Find AB (km, 1 d.p.).

4. From P, A is 7 km on bearing 020° and B is 9 km on bearing 080°. Find AB (km, 1 d.p.).

5. Triangle PQR has PQ = 15 km, PR = 20 km, angle QPR = 65°. Find QR (km, 1 d.p.).

6. From O, X is on bearing 030° and Y is on bearing 090°. OX = 10 km, OY = 10 km. Find XY (km, 1 d.p.).

7. Triangle ABC: AB = 8 km, BC = 11 km, angle ABC = 55°. Find AC (km, 1 d.p.).

8. From lighthouse L, ship X is 20 km on bearing 070°, ship Y is 15 km on bearing 130°. Find XY (km, 1 d.p.).

Practice — 25 Mixed Questions

All answers in degrees (°) or km as indicated.

1. Write the bearing of East as 3 digits (°).

2. Bearing of B from A is 095°. Find bearing of A from B (°).

3. Bearing of X from Y is 210°. Find bearing of Y from X (°).

4. Write South-West as a bearing (°).

5. From P, bearings to A and B are 040° and 115°. Find angle APB (°).

6. Bearing of Q from P is 175°. Find bearing of P from Q (°).

7. A walks 5 km North and 5 km East. Find bearing of final position from start (° to nearest degree).

8. Ship sails bearing 120° for 30 km. How far East? (km, 1 d.p.)

9. Ship sails bearing 120° for 30 km. How far South? (km, 1 d.p.)

10. Bearing of C from D is 300°. Find bearing of D from C (°).

11. From H, A is 10 km on bearing 050° and B is 10 km on bearing 110°. Find AB (km, 1 d.p.).

12. Write bearing 7° as a three-figure bearing (°).

13. Bearing of R from S is 135°. Find bearing of S from R (°).

14. A town is 3 km East and 4 km North of a village. Find the bearing of the town from the village (° to nearest degree).

15. From P, A is on bearing 030°, B is on bearing 150°. PA = PB = 20 km. Find AB (km, 1 d.p.).

16. A plane flies bearing 270° for 80 km. How far West is it? (km)

17. Bearing of B from A is 015°. Find bearing of A from B (°).

18. From X, Y is 5 km on bearing 060°. How far North is Y from X? (km, 1 d.p.)

19. Write North-West as a bearing (°).

20. From H, bearings to A and B are 075° and 165°. Angle AHB = ? (°)

21. A hiker walks 8 km due East. What is the bearing of the start from their current position? (°)

22. From P, A is 6 km on bearing 000° and B is 6 km on bearing 090°. Find AB (km, 1 d.p.).

23. Bearing of G from F is 248°. Find bearing of F from G (°).

24. From A, B is 15 km on bearing 060° and C is 20 km on bearing 120°. Find BC (km, 1 d.p.).

25. A ship sails 10 km on bearing 045° then 10 km on bearing 135°. How far South of the start is the ship? (km, 1 d.p.)

Challenge — 12 Questions (IGCSE Extended Level)

1. From P, A is on bearing 035° and B is on bearing 305°. PA = PB = 12 km. Find AB (km, 1 d.p.).

2. A ship sails 50 km on bearing 030°, then 40 km on bearing 120°. Find the distance from the start to the final position (km, 1 d.p.).

3. In triangle PQR, PQ = 10 km on bearing 060°, QR = 14 km on bearing 150° from Q. Find PR (km, 1 d.p.).

4. A is 12 km due North of C. B is on a bearing of 065° from A and on a bearing of 020° from C. Find AB (km, 1 d.p.).

5. From lighthouse L, wreck W is 8 km on bearing 070°. From port P (due East of L), W is on bearing 320°. LP = 10 km. Find LW using the sine rule (km, 1 d.p.) — check answer: 8 km.

6. A helicopter flies 80 km on bearing 040°, then turns and flies 60 km on bearing 160°. Find the bearing of the final position from the start (° to nearest degree).

7. From town T, village V is 25 km on bearing 115°. Another town U is 18 km from V on bearing 205° from V. Find UV's bearing from T (° to nearest degree).

8. Three points A, B, C form a triangle. AB = 9 km, AC = 11 km, angle BAC = 48°. Find BC (km, 1 d.p.).

9. In triangle PQR, PQ = 20 km, QR = 15 km, PR = 18 km. Find the angle QPR (° to nearest degree).

10. A and B are 50 km apart. The bearing of B from A is 060°. C is 35 km from A on bearing 120°. Find BC (km, 1 d.p.).

11. A plane flies 200 km on bearing 025°. It then flies 300 km on bearing 115°. Find the total displacement (distance from start, km to 1 d.p.) and bearing of final position from start (° to nearest degree). Enter the distance only.

12. From tower T, mast A is 5 km on bearing 020° and mast B is 7 km on bearing 080°. Find the bearing of B from A (° to nearest degree).

Exam Style Questions

Mark-scheme style. Show working in your book. Enter final answers for self-marking.

Question 1 — Bearings and Back Bearings [4 marks]

Ship S leaves port P on a bearing of 125°.

(a) Write down the bearing of P from S. [1]

(b) A lighthouse L is on a bearing of 035° from P. The bearing of L from S is 305°. Are P, S and L collinear? Enter 1 for yes, 0 for no. [1]

(c) Another ship Q sails from P on the back bearing of 125°. What bearing does Q sail on? [1]

(d) Give the bearing that corresponds to sailing due North-West. [1]

Question 2 — Calculating Bearings Using Alternate Angles [4 marks]

Town B is due South of town A. The bearing of town C from town A is 072°.

(a) Find the bearing of C from B (alternate angles). [2]

(b) If D is due East of B, and the bearing of C from B is 072°, what is the angle CBD? [2]

Question 3 — Right-Angled Trigonometry [5 marks]

A boat travels from port P due East for 9 km to point Q, then due North for 12 km to point R.

(a) Find the bearing of R from P (° to nearest degree). [3]

(b) Find the straight-line distance PR (km, 1 d.p.). [2]

Question 4 — Cosine Rule with Bearings [5 marks]

From harbour H, ship A is 24 km on bearing 055° and ship B is 18 km on bearing 145°.

(a) Find the angle AHB. [1]

(b) Calculate the distance AB (km, 1 d.p.). [3]

(c) Given that angle HAB = 58.2°, find the bearing of B from A (° to nearest degree). [2]

Question 5 — Multi-Step Bearing and Sine Rule [6 marks]

Three radio masts P, Q and R are positioned as follows: Q is 30 km from P on bearing 065°. R is 45 km from P on bearing 155°.

(a) Write down the angle QPR. [1]

(b) Calculate QR (km, 1 d.p.). [3]

(c) Using the sine rule, find angle PQR (° to 1 d.p.). [2]

Bearings IGCSE Extended