Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 The bearing of B from A is 052°. What is the compass bearing of A from B?
Show worked solution
Back bearing: 052° + 180° = 232°. That is 52° past south, towards west: S52°W. N52°E is the bearing of B from A, not the way back; S52°E turns towards east instead of west; S38°W measures 38° from west (270° − 232°) instead of from south.
2 Find the true bearing of B from A, and of A from B.
Show worked solution
B is S40°E from A, so the bearing is 180° − 40° = 140°. The back bearing is 140° + 180° = 320°.
3 A hiker walks 10 km due south and then 24 km due east. How far is she from her starting point, in km?
Show worked solution
d2 = 102 + 242 = 100 + 576 = 676, so d = √676 = 26 km. (Adding the legs gives 34 km, the distance walked.)
Level 2 · Core
4 A drone flies 5 km due north and then 5 km due west. Which gives the bearing and distance of the drone from its starting point?
Show worked solution
The legs are equal, so the angle at the start is 45°, on the west side of north: N45°W = 360° − 45° = 315°. The distance is √(25 + 25) = √50 = 5√2 km. So 315°, 5√2 km. 045° forgets the west; 10 km adds the legs; 135° faces south-east.
5 The compass bearing of B from A is N65°W. Find the true bearing of B from A, and the true bearing of A from B.
Show worked solution
N65°W = 360° − 65° = 295°. The back bearing is 295° − 180° = 115°, which is S65°E.
6 A ship sails 10 km on a bearing of 030°, then 10√3 km on a bearing of 120°. How far is it from its starting point, in km?
Show worked solution
120° − 30° = 90°, so the turn is a right angle and the two legs are the short sides of a right-angled triangle. d2 = 102 + (10√3)2 = 100 + 300 = 400, so d = 20 km.
Level 3 · Challenge
7 Three statements:
1 If B is on a bearing of 200° from A, then A is on a bearing of 020° from B.
2 The directions 070° and 160° are at right angles.
3 The back bearing of N20°E is S20°E.
Which statements are correct?
1 If B is on a bearing of 200° from A, then A is on a bearing of 020° from B.
2 The directions 070° and 160° are at right angles.
3 The back bearing of N20°E is S20°E.
Which statements are correct?
Show worked solution
200° − 180° = 020° (1 ✓). 160° − 70° = 90° (2 ✓). N20°E = 020°, and 020° + 180° = 200° = S20°W, not S20°E (3 ✗). 1 and 2.
8 A plane flies 9√3 km due west and then 9 km due south. Find its distance and its true bearing from the starting point.
Show worked solution
d2 = (9√3)2 + 92 = 243 + 81 = 324, so d = 18 km. At the start, the angle from south: tan θ = 9√3 ÷ 9 = √3, so θ = 60° towards west: S60°W = 180° + 60° = 240°.
9 Port B is 11 km due east of port A. A lighthouse L is due north of A, and on a bearing of 330° from B. How far is the lighthouse from B, in km?
Show worked solution
At B, the bearing 330° points 30° west of north, so angle ABL = 90° − 30° = 60°. In triangle ABL (right angle at A), BA = 11 is adjacent to that angle and BL is the hypotenuse: cos 60° = 11 ÷ BL, so BL = 11 ÷ 1/2 = 22 km.