Quantitative Aptitude · Direction Tests

Pythagoras Theorem in Direction Tests

Updated 1 October 2026 · Fact-checked

In direction tests, a person often walks in perpendicular directions, such as north then east. The shortest distance from start to end is the hypotenuse of a right-angled triangle. Add the net north-south and net east-west distances separately, then use √(a² + b²), or spot a triplet like 3-4-5.

Understand Pythagoras Theorem in Direction Tests

Direction questions describe a walk: go 3 km north, turn right, go 4 km, and so on. The question then asks, "How far is he from the starting point?" This is the shortest distance, a straight line from start to end. It is not the total distance walked.

North and east are at 90° to each other. So are north and west, south and east, and south and west. When a path has turns of 90°, the start point, the corner and the end point form a right-angled triangle. The straight line from start to end is the hypotenuse, the side opposite the right angle.

Pythagoras theorem gives its length. If the two perpendicular sides are a and b, the hypotenuse is √(a² + b²). Before using it, combine the movements. Walking 10 km north and then 4 km south leaves a net of 6 km north. Do this separately for the north-south line and the east-west line.

Exam numbers are usually chosen to form Pythagorean triplets, so the answer is a whole number. If you know 3-4-5, 5-12-13, 8-15-17 and 7-24-25, you can write the answer without squaring or taking roots. Multiples also work, so 6-8-10 and 9-12-15 come from 3-4-5.

Key formulas to remember

Pythagoras theorem: Shortest distance = √(a² + b²); a and b are the net perpendicular displacements, one along north-south and one along east-west. Valid only when the two are at right angles.
Net displacement on one line: Net = (movement one way) − (movement the opposite way); Do this separately for north-south and east-west. Ignore the sign for the final squares.
Common triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29; Any multiple of a triplet is also a triplet, such as 6-8-10, 9-12-15, 10-24-26, 15-20-25.
Finding a missing leg: a = √(c² − b²); Use when the hypotenuse c and one leg b are given.
Total distance vs shortest distance: Total distance = sum of all legs walked; Shortest distance is never more than the total distance walked. They are equal only if the whole walk is in one straight line.

How to solve Pythagoras Theorem in Direction Tests questions

Use this method for any question that asks for the distance or shortest distance between start and end points.

1Draw a quick sketch. Mark the start point and put north at the top.
2Plot each leg in order, turning left or right correctly from the current facing direction.
3Add the movements along north-south, subtracting opposite directions, to get one net value.
4Do the same for east-west to get the second net value.
5If one net value is zero, the answer is simply the other value. The person is in a straight line from the start.
6If both are non-zero, they are the two legs of a right triangle. Check whether they fit a triplet or a multiple of one.
7If not, compute √(a² + b²). Match the result to the options.
8Check that the answer is not more than the total distance walked.

Quickest way: Triplet spotting with option elimination

When to use it: Use it when the two net legs are whole numbers and the options are whole numbers, which is common in MCQs.

After netting, look at the two numbers. Ask: is this 3-4, 5-12, 8-15, 7-24 or a multiple?
If yes, write the hypotenuse directly: 3-4 gives 5, 5-12 gives 13, 8-15 gives 17, 7-24 gives 25.
For multiples, scale up. Legs 15 and 20 are 5 times 3 and 4, so the answer is 25.
If not a triplet, find the nearest squares. For legs 6 and 7, 36 + 49 = 85, so the answer is a little over 9. Pick the closest option.
Eliminate any option larger than the total distance walked, or smaller than the bigger leg. The hypotenuse must exceed each leg.
If a question needs a very long sketch and you are unsure of a turn, skip it. A wrong answer costs 0.25 marks.

Common mistakes in Pythagoras Theorem in Direction Tests

Giving the total distance walked as the shortest distance.
Students add all the legs because the numbers are already on the page.
Fix: Read the question word. "Shortest distance" or "how far from start" means a straight line. Use Pythagoras.
Applying Pythagoras to the raw legs without netting opposite movements.
Students treat every leg as a side of the triangle.
Fix: First combine north with south and east with west. Only the two net values form the triangle.
Adding instead of squaring: using a + b or √(a + b).
Rushing under time pressure.
Fix: Write a² + b² before the root. Check with a triplet: 3 and 4 must give 5, not 7.
Turning the wrong way after a left or right turn.
A turn is taken relative to the current facing direction, not to north.
Fix: Facing north, right is east. Facing east, right is south. Facing south, right is west. Facing west, right is north. Left is the reverse. Update the facing direction after every turn.
Choosing a hypotenuse that is smaller than one of the legs.
Mixing up which side is the longest.
Fix: The hypotenuse is always the longest side of the triangle. Reject options that are not larger than both legs.
Not recognising scaled triplets, such as 24-32.
Students only memorise the base triplets.
Fix: Divide both legs by their common factor. 24 and 32 divide by 8 to give 3 and 4, so the hypotenuse is 8 × 5 = 40.

Worked examples

Example 1

A man walks 9 km north, then turns around and walks 4 km south, and then turns left and walks 12 km. How far is he from the starting point? (A) 13 km (B) 15 km (C) 17 km (D) 21 km

Show the solution

Start facing north. Walk 9 km north.
Turn around to face south. Walk 4 km south.
Turn left. Facing south, left is east. Walk 12 km east.
North-south net: 9 km north − 4 km south = 5 km north.
East-west net: 12 km east.
Legs 5 and 12 form a triplet, so the hypotenuse is 13 km.
Check: total distance walked is 9 + 4 + 12 = 25 km, and 13 km is less than that.

Answer: (A) 13 km

Example 2

Rohan walks 8 km west, then turns left and walks 6 km. He then turns left and walks 3 km, and finally turns right and walks 6 km. How far is he from the starting point? (A) 9 km (B) 10 km (C) 13 km (D) 17 km

Show the solution

Start facing west. Walk 8 km west.
Turn left. Facing west, left is south. Walk 6 km south.
Turn left. Facing south, left is east. Walk 3 km east.
Turn right. Facing east, right is south. Walk 6 km south.
East-west net: 8 km west − 3 km east = 5 km west.
North-south net: 6 km south + 6 km south = 12 km south.
Legs 5 and 12 form a triplet, so the hypotenuse is 13 km.

Answer: (C) 13 km

Example 3

A girl starts from her house and walks 20 m east, then 15 m north, then 10 m west, then 15 m south. How far is she from the house? (A) 5 m (B) 10 m (C) 20 m (D) 25 m

Show the solution

East-west: 20 m east − 10 m west = 10 m east.
North-south: 15 m north − 15 m south = 0.
One net value is zero, so she is on a straight line from the house.
Distance = 10 m. No Pythagoras is needed.

Answer: (B) 10 m

Exam tips

Net the opposite directions first. Many questions are built so that one axis cancels to zero, which saves a square root.
Memorise 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and scale them. Examiners choose numbers to fit these.
Always update your facing direction after each turn and write it next to the leg in your sketch.
Check what is asked: shortest distance, total distance, or direction. Some questions ask for both, so read to the end.
With 0.25 negative marking, skip a long walk if your sketch has a doubtful turn, and come back if time remains.

Practice questions from Direction Tests

Pythagoras Theorem in Direction Tests: frequently asked questions

When do I use Pythagoras theorem in direction tests?

Use it when the net displacement has both a north-south part and an east-west part, both non-zero. The two parts are at right angles, so the shortest distance is the hypotenuse. If one part is zero, the answer is just the other part.

Which Pythagorean triplets should I remember for CA Foundation?

Remember 3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus their multiples such as 6-8-10 and 9-12-15. These cover most direction test questions in objective papers.

Is shortest distance the same as total distance walked?

No. Total distance is the sum of every leg walked. Shortest distance is the straight line from the start to the end. It is equal to the total only when the whole path is a straight line.

What if the answer is not a perfect square root?

Find the two nearest perfect squares and estimate. For example, √65 lies between 8 and 9 and is just over 8, about 8.06. Then pick the option closest to that value, after rejecting options that break the hypotenuse rule.