1. Dr J Frost (jfrost@tiffin.kingston.sch.uk)
GCSE :: Bearings
Dr J Frost
Objectives: (a) Measure bearings using a protractor. (b) Construct bearings, involving map scales (c) Solve bearings problems using given angles.
Last modified: 28th January 2018

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3. Starter Puzzle Q
Here some aerial pictures of various airport runways Dr Frost has been on. Can you work out what the numbers mean?
Beijing Capital International Airport
Heathrow - London
JFK – New York
Solution: If we add a 0 on the end, we get the angle clockwise from North the runway faces!
Portland International Airport - Oregon
Rhodes Airport - Greece
Hilo Airport - Hawaii
EuroAirport – Basel-Mulhouse-Freiberg

4. Bearings
We understand an ‘angle’ to mean ‘the amount of turn’. Bearings are more specifically the amount of turn clockwise, starting from North.
! A bearing is an angle measured clockwise from North. We use 3 digits.
𝟎𝟎𝟎°
Historical reason for 3 digits?
Bearings used to be transmitted digit by digit from ship to ship. Were some bearings two digits, it would be ambiguous whether two digits was intended or a digit was lost in transmission. ? E B
𝟐𝟕𝟎°
𝟎𝟗𝟎° Q Q
𝟏𝟖𝟎°
Bearings can be thought of as a numerical version of cardinal directions (North, West, etc)
The bearing of: A Q C Q
B from A
000
A from B
180
C from A
090
D from A
225
A from D
045
A from E
135
E from A
315 ? ? ? ? D Q ? ? ?

5. Measuring Bearings ? 056° 180°+56°=236°
Fro Tip: You’re measure the angle at whatever appears just after the word ‘from’.
Measure the bearing of 𝐵 from 𝐴.
Click to Fromeasure > N
056°
Measure the bearing of 𝐴 from 𝐵. 𝐵
180°
Our protractor only goes up to 180° so start measuring from South.
+56°
Click to Fromeasure > N
180°+56°=236°
What do you notice about the two bearings? 𝐴
! If we swap “A from B” with “B from A”, we add or subtract 180° to get the bearing. ?
Position your protractor so the centre is on the point you’re measuring (whatever appears after the word ‘from’).
Ensure the 0 on your protractor lines up with the North arrow.

6. Test Your Understanding
Question on provided sheet. 𝑁 𝐴 𝑁 𝐶 𝑁 𝐵
B from A
116
A from B
296
C from B
073
D from C
228
A from D
334
D from A
154 ? ? 𝑁 ? 𝐷 ? ? ?

7. Continued on next slide…
Exercise 1 – Measuring Bearings
(On supplied sheet) 1
In each diagram determine: (without a protractor)
(i) the bearing of B from A and (ii) the bearing of A from B. a b 𝐴 c 𝐴 𝐵 𝐵 𝐴
𝟎𝟗𝟎° 𝟐𝟕𝟎°
𝟏𝟖𝟎° 𝟎𝟎𝟎°
𝟐𝟕𝟎° 𝟎𝟗𝟎°
𝟎𝟒𝟓° 𝟐𝟐𝟓°
𝟏𝟑𝟓° 𝟑𝟏𝟓°
? a 𝐵
? b
? c d e 𝐵 𝐴
? d
? e 𝐴 𝐵
At what bearing is the sun if I read the following off my compass:
(a) W 𝟐𝟕𝟎° (b) E 𝟎𝟗𝟎° (c) NE 𝟎𝟒𝟓°
(d) SW 𝟐𝟐𝟓° (e) SE 𝟏𝟑𝟓° (f) NEE 𝟎𝟔𝟕.𝟓° 2 ? ? ? ? ? ?
Continued on next slide…

8. Exercise 1 – Measuring Bearings
(On supplied sheet) 3
[Edexcel GCSE Nov2015-1H Q11a] Manchester airport is on a bearing of 330° from a London airport. Find the bearing of the London airport from Manchester airport.
𝟏𝟓𝟎°
[Edexcel GCSE Jun2015-2H Q13] Martin and Janet are in an orienteering race. Martin runs from checkpoint A to checkpoint B, on a bearing of 065°. Janet is going to run from checkpoint B to checkpoint A.
Work out the bearing of A from B.
𝟐𝟒𝟓° 7
Determine the bearing of:
Cambridge from London. 𝟎𝟔𝟔°
London from Cambridge. 𝟐𝟒𝟔° N
Cambridge ? N
London ? ? 4 8
Determine the bearing of:
Hamburg from Munich. 𝟏𝟏𝟒°
Munich from Hamburg. 𝟐𝟗𝟒° N
Munich ? N
Hamburg ? 5 N ?
Argos
Determine the bearing of:
Lidl from Argos. 𝟐𝟎𝟖°
Argos from Lidl. 𝟎𝟐𝟖° ? N ?
Lidl 6 N
Determine the bearing of:
B from A. 𝟏𝟓𝟎°
A from C. 𝟐𝟗𝟏°
B from C. 𝟐𝟒𝟖° A ? N C ? N ? B

9. You have a map of Kingston-upon-Thames…
Scale 1:300
 What does this scale mean? ?
Real life distances are 300 times larger than map distances. e.g. 1cm on the map represents 300cm in real life.
75cm
Tiffin
John Lewis
What distance is this in real life?
75×300=22500 cm =225m ?

10. Further Examples
A map scale is 1 : A distance is measured on a map of 4cm. What does this represent in real life? ?
4 cm ×20 000= cm
=800 m
When scaling, be consistent with the unit used.
A distance on a map of 5cm represents 2km in real life. Determine the map scale in the form 1:𝑛 . ?
Write the ratio of the two distances (with units), and convert one distance so that the units are the same.
5 cm : 2 km
5 cm : 2000 m
5 cm : cm
1 :

11. Test Your Understanding
[Edexcel IGCSE Nov H Q8]
The scale of a map is 1 : On the map, the distance between two schools is 19.6 cm.
Work out the real distance between the schools.  Give your answer in kilometres. ?
19.6 cm ×𝟓𝟎 𝟎𝟎𝟎=𝟗𝟖𝟎 𝟎𝟎𝟎 cm
= 9800 m
= 9.8 km
[Edexcel GCSE(9-1) Mock Set 2 Spring F Q10]
A map has a scale of 1 cm to 25 km. The distance between the cities of Edinburgh and Bristol is 500 km. What is the distance on the map between these two cities? ?
𝟓𝟎𝟎 𝟐𝟓 =𝟐𝟎 cm

12. Click to start Fromanimation >
Constructing Maps
You can combine your knowledge of map scales with that of bearings in order to plot locations on a map.
Bob is at a bearing of 070° from Alice, 6 km away. Using a scale of 1cm : 1km, mark where Bob is. N
Bob!
Use your pencil to mark the correct angle.
Alice
Click to start Fromanimation >

13. Click to start Fromanimation >
Further Example
You can combine your knowledge of map scales with that of bearings in order to plot locations on a map.
Chelsea is at a bearing of 200° from Tarquin, 40 km away. Using a scale of 1cm : 5km, mark where Bob is.
Click to start Fromanimation > N
180°
Tarquin
200=180+20, so we start measure clockwise from South.
Chelsea

14. Exercise 2 – Constructing Bearings
(On supplied sheet)
𝐵 is at a bearing of 30° from 𝐴, 5cm away. Mark the position of 𝐵. 3
𝐹 is at a bearing of 250° from 𝐸, 4.1 cm away. Mark the position of 𝐹. 1 ? ? 𝐵 N
5cm N 𝐸 𝐴 𝐹
4.1cm 2
𝐷 is at a bearing of 135° from 𝐶, 6.2cm away. Mark the position of 𝐷. 4
A ship is at a bearing of 70° from a ship, 6 km away. Clapp Island is at a bearing of 150° from the ship, 4 km away. By first plotting the locations of the ship and island (using a scale of 1cm : 1km), estimate how far away the island is from the lighthouse. ? N 𝐶 ? N
6.2cm 𝑆 𝐷
Distance: 7.8km
4 km 𝐿
6 km 𝐶𝐼

15. Exercise 2 – Constructing Bearings
(On supplied sheet)
Anne is 3m away from Ishan at a bearing of 100°. Katniss is 7m away from Anne at a bearing of 260°. By plotting the positions of Anne and Katniss (using a scale of 1cm : 1m), estimate the distance of Katniss from Ishan.
A map scale is 1:2000.
The distance on a map between two locations is 3.5cm. What real distance does this represent? 70 metres
A real life distance is 50km. What distance would this be on a on a large map? Give your answer in metres. 25 metres
A map scale is 1: On a map two locations are 6.4cm apart. How far apart are they in real life? Give your answer in kilometres km
A map distance of 8cm represents 40km in real life. Determine the map scale in the form 1:𝑛 𝟏:𝟓𝟎𝟎 𝟎𝟎𝟎 5 7 ? ? N 𝐼
3 m
Distance: 4.3m 𝐴 ? 𝐾
7 m 8
Ron is at a bearing of 300° from Harry at a distance of 500m. Hermione is at a bearing of 70° from Ron at a distance of 650m. But using a scale of 1cm : 100m, determine the distance of Hermione from Harry. 6 ? 9 ?
6.5 m 𝐻𝑒 ? 𝑅𝑜 N
5 m 𝐻𝑎

16. Bearings calculations using known angles
Often in questions, angles will already be given to you. You need to use your knowledge of angle laws to calculate bearings.
Recap: 𝑥
180−𝑥 ?
Alternate angles are equal.
Corresponding angles are equal. ? ?
Cointerior angles add to 180°.
Angles around a point sum to 360°. ?

17. Examples
[Edexcel GCSE June2010-2F Q17] Work out the bearing of 𝐵 from 𝐴.
Always start by drawing the bearing in. Recall that we go clockwise from North and we draw the angle at the point after the word ‘from’.
𝟏𝟖𝟎°+𝟒𝟎°=𝟐𝟐𝟎° N
90°
Determine the bearing of 𝐴 from 𝐵. 𝐵
55°
𝟗𝟎°+𝟓𝟓°=𝟏𝟒𝟓°
35° 𝐴

18. Harder Example
The bearing of 𝐵 from 𝐴 is 070° and the bearing of 𝐶 from 𝐵 is 100°. The distance from 𝐴 to 𝐵 is the same as 𝐵 to 𝐶.
Determine the bearing of 𝐴 from 𝐶.
Note that when we draw the diagram, we need not draw the angles accurately, because we are using angle laws, and not a protractor, to determine the answer.
Cointerior angles add to 180°
Put angles given in question on diagram. 𝐵
100°
We’re told 𝐴𝐵 and 𝐵𝐶 are the same length.
110°
70°
150°
15° 𝐴
We now have enough angles to work out the bearing of 𝐴 from 𝐶.
80°
15° 𝐶
Angles around a point add to 360°.
𝟏𝟔𝟓°
So triangle 𝐴𝐵𝐶 is isosceles.

19. Test Your Understanding1 𝑁 𝐵
Determine the bearing of 𝐵 from 𝐴.
30° 𝐴 ?
𝟑𝟎𝟎° 2
In the diagram, 𝐴𝐵=𝐴𝐶. Determine the bearing of:
𝐶 from 𝐴.
𝐶 from 𝐵 𝑁 𝑁
104° 𝐵 ? 𝐴
𝟏𝟐𝟔°
𝟏𝟗𝟏°
50° ? 𝐶

20. Exercise 3 – Bearings using angle laws
(On supplied sheet)
Determine the bearing of 𝐵 from 𝐴.
Determine the bearing of 𝐵 from 𝐴.
In the diagram, 𝐴𝐵=𝐴𝐶. Determine the bearing of:
𝐵 from 𝐶. 𝟕𝟎°
𝐶 from 𝐵. 𝟐𝟓𝟎°
𝐵 from 𝐴. 𝟏𝟒𝟎° 1 2 4 𝑁 ? 𝑁 𝑁 ? 𝐴 ? 𝐴 𝐴
50°
70°
25° 𝐵 𝐵
Solution:𝟐𝟐𝟎° ? 𝐵 𝐶 ?
Solution:𝟏𝟓𝟓°
Determine the bearing of 𝐵 from 𝐴. 3 5 𝐵
In the diagram 𝐴𝐶=𝐵𝐶. Determine the bearing of:
(a) 𝐵 from 𝐴 𝟓𝟓°
(b) 𝐶 from 𝐴.
𝟏𝟏𝟎° 𝑁 𝐴 𝐵
70° 𝑁
130° 𝐶 ? 𝐴 ? ?
Solution: 𝟒𝟎°

21. Exercise 3 – Bearings using angle laws
(On supplied sheet)
[Edexcel IGCSE Jan2014(R)-4H Q10b] The diagram shows the positions of a yacht Y, a ship S and a beacon B. The bearing of B from Y is 228°.
The bearing of S from Y is 118°.
(a) Find the size of the angle BYS.
(b) Given also that BY = SY, find the bearing of S from B.
[Edexcel IGCSE Nov2009-4H Q3b] The bearing of B from A is 062°.  C is due south of B.
AB = CB.
Work out the bearing of C from A. 6 7 ?
𝟏𝟏𝟎°
𝟎𝟖𝟑° ? ?
𝟏𝟐𝟏°

22. Exercise 3 – Bearings using angle laws
(On supplied sheet)
[OCR GCSE(9-1) Nov H Q9]
The diagram shows the positions of two towns, Amton and Bisham.
The bearing of Bisham from Amton is 𝑏°. The bearing of Amton from Bisham is 6𝑏°.
Calculate the 3-figure bearing of Amton from Bisham.
The bearing of 𝐵 from 𝐴 is 96° and the bearing of 𝐶 from 𝐵 is 104°. The distance from 𝐴 to 𝐵 is the same as 𝐵 to 𝐶. Determine the bearing of 𝐶 from 𝐴.
Solution: 𝟏𝟎𝟎° 8 9 ? ?
Solution: 𝟐𝟏𝟔°