X is due east point of y on a coast. Z is another point on the coast but 6.0km due south of Y. If the distance ZX is 12km, calculate the bearing of Z from X

Analyze the question

X is due east of Y, and Z is 6 km due south of Y. So the angle at Y is 90°, and triangle XYZ is right-angled at Y.

• YZ = 6 km
• ZX = 12 km (the hypotenuse)

We want the bearing of Z from X. This is measured clockwise from North at X.

Concept

Bearings are measured clockwise from the North line at the point you are standing on. Since we want the bearing of Z from X, we stand at X.

Solution

Set coordinates with Y at the origin: Y = (0, 0), X = (d, 0), Z = (0, -6).

Find XY using Pythagoras: XY² = 12² - 6² = 144 - 36 = 108, so XY = 6√3 km.

Now find the angle at X in triangle XYZ: sin(∠YXZ) = YZ / XZ = 6/12 = 1/2 so ∠YXZ = 30°.

From X, Y lies due west (bearing 270°). Z lies south-west of X, below the west line. The direction XZ is 30° south of west, which is the same as 30° beyond 270° going anticlockwise from west toward south.

Going clockwise from North: North is 0°, East is 90°, South is 180°, West is 270°. A direction 30° south of west lies between South (180°) and West (270°), at 270° - 30° = 240°.

So the bearing of Z from X is 240°.

Check: The back bearing of X from Z would be 240° - 180° = 60°. From Z, X lies north-east, 60° east of north. Check: tan of the angle at Z = XY/YZ = 6√3/6 = √3, so the angle is 60° from north. ✓.

Answer: A. 240°

Tip: For bearing questions, always draw the North line at the point you are standing on (the "from" point). Then measure clockwise. A quick check: the bearing of A from B and the bearing of B from A always differ by 180°.