Competitive exam reasoning

Direction and Distance Shortcuts for Reasoning

Coordinates turn a long walking route into two totals. Track east-west movement on one axis and north-south movement on the other, while treating the traveller's facing direction separately.

1

Fix a coordinate system

Use east as positive x, west as negative x, north as positive y and south as negative y. Start from (0, 0).

Solved exampleFour kilometres east and three north ends at (4, 3), northeast of the start.
Common trapThe page compass stays fixed even when the traveller turns.
2

Cancel opposite movements

Add horizontal movements together and vertical movements together. Opposite directions subtract from each other.

Solved example8 east, 3 west, 6 north and 2 south leaves a displacement of 5 east and 4 north.
Common trapDisplacement is not the same as the total path travelled.
3

Use right-triangle distance

Once net horizontal and vertical components are known, apply √(x² + y²) for the shortest distance.

Solved exampleNet movements of 6 and 8 give a shortest distance of 10.
Common trapUse net components, not every leg in the route.
4

Memorise common triples

Recognise 3-4-5, 5-12-13 and 8-15-17, including scaled versions, to avoid square-root calculation.

Solved exampleA 9-by-12 displacement is three times 3-4-5, so its straight-line distance is 15.
Common trapA triple applies only when the two components are perpendicular.
5

Represent turns as a cycle

Write N, E, S, W clockwise. Move one position forward for a right turn and one backward for a left turn.

Solved exampleStarting north, right-right-left ends facing east.
Common trapFinal facing and final location relative to the start are different answers.

Frequently asked questions

What is the difference between distance and displacement?

Distance is the full route travelled. Displacement is the shortest straight-line change from the starting point to the ending point.

Which Pythagorean triples are most useful for exams?

The most common are 3-4-5, 5-12-13 and 8-15-17, along with multiples such as 6-8-10 and 9-12-15.