Direction & Distance

medium importance~1-2 Q in Tier 18 formulas⚡ 6 shortcuts4 subtopics

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Direction conventions and displacement

⏱ 5 min read🧩 6 question types🎯 13 practice Q

The idea in one minute

A direction question is a walk on a plain map. Put North at the top and East on the right. Turn every sentence into one small move, then find the straight line from start to finish. That line is the shortest distance, and it is not the same as the distance the person walked.

The four directions

Draw a map with North at the top and East on the right. South is at the bottom. West is on the left.

Between them sit four mixed directions: North-East, North-West, South-East and South-West. "Towards the sunrise" means East. "Towards the sunset" means West.

Rule: Draw the map once, with North up. Never turn the page when the walker turns.

Left and right belong to the walker

A left turn does not mean "West". It means "to the left of the way he faces now".

The walker faces	Left is	Right is
North	West	East
South	East	West
East	North	South
West	South	North

A man walks North and turns left. He now walks West. The same left turn from South would send him East.

Watch: "Left = West" is true only when the walker faces North. Check the table for every turn.

Turn each sentence into a move

Start at the middle of the map. Each sentence moves you. Keep a running tally on two lines: East-West and North-South.

East and North count as plus.
West and South count as minus.

Example: "8 km East, 6 km West, then 3 km East." The East-West line is $8 - 6 + 3 = 5$ . He is 5 km East of the start.

Cancel opposite legs first

A leg is one straight part of the walk. Two legs that go opposite ways on the same line cancel each other.

A man walks 15 km North, 9 km East, then 15 km South. The 15 North and 15 South cancel. Only 9 km East is left.

Tip: Cancel first, then calculate. Most questions hide two or three cancellations.

Shortest distance

The shortest distance is the straight line from start to finish. After cancelling, you have a net East move and a net North move. They are the two short sides of a right triangle.

d = \sqrt{(\text{net East})^2 + (\text{net North})^2}

Example: 3 km East and 4 km North gives $\sqrt{9 + 16} = 5$ km.

Most exam answers come from these number triples:

Short sides	Straight line
3 and 4	5
5 and 12	13
8 and 15	17
7 and 24	25

Doubles and triples also work: 6 and 8 give 10, and 9 and 12 give 15.

Diagonal legs

A leg such as "5 $\sqrt{2}$ km North-East" is a slanting step. It moves 5 km North and 5 km East at the same time.

North-East: $k$ North and $k$ East.
North-West: $k$ North and $k$ West.
South-East: $k$ South and $k$ East.
South-West: $k$ South and $k$ West.

Example: 5 $\sqrt{2}$ km North-East, then 5 $\sqrt{2}$ km North-West. The East and West parts cancel. The North parts add up to 10 km.

Distance walked or shortest distance

Read the last line twice. Each phrase asks for a different number.

The question asks	Find
How far is he from the start?	Straight line (Pythagoras)
How much distance did he walk?	Add all the legs
How much shorter is the direct path?	Total walked minus straight line
In which direction is he from the start?	Compass word of the net moves

Watch: The walked distance and the straight line are always two different options. Do not pick the wrong one.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Straight line after two perpendicular legs

How to spot it:

The legs run North-South and East-West. The question asks how far he is from the starting point.

d = \sqrt{(\text{net East})^2 + (\text{net North})^2}

Method

Write each leg as an East-West move or a North-South move.
Cancel opposite legs and find the net East and net North.
Look for a number triple (3-4-5, 5-12-13, 8-15-17).
Otherwise apply the square root formula.

Why it works:

The two net moves are the short sides of a right triangle. The straight line home is its longest side.

Try this

A man walks 9 km East and then 12 km North. How far is he from the starting point?

Show solution

Net moves: 9 km East and 12 km North.
9 and 12 are 3 times 3 and 3 times 4.
Distance = $3 \times 5 = 15$ km.

Answer

15 km

Type 2common2 practice Q

Legs that cancel out

How to spot it:

The path doubles back. A leg North is followed by a leg South, or East by West.

Method

Pair each leg with the leg that goes the opposite way.
Cancel the pairs. Subtract the smaller from the larger.
Use only what is left to find the distance.

Why it works:

Only the start and the end matter. Retraced legs add nothing to the gap.

Try this

A cyclist rides 15 km West, 6 km North, 15 km East and then 2 km North. How far is he from the starting point?

Show solution

15 km West and 15 km East cancel.
North legs add up: $6 + 2 = 8$ km.
Only 8 km North is left.

Answer

8 km

Type 3common2 practice Q

Diagonal legs written with root 2

How to spot it:

A leg is written as $k\sqrt{2}$ km towards North-East, North-West, South-East or South-West.

k\sqrt{2}\ \text{diagonal} = k\ \text{on each of two sides}

Method

Split each diagonal leg into two equal parts of $k$ .
Add up the East-West parts and the North-South parts.
Find the straight line from the net moves.

Why it works:

A diagonal step of $k\sqrt{2}$ is the slanting side of a right triangle with two sides of $k$ .

Try this

A man walks $8\sqrt{2}$ km North-East and then 2 km West. How far is he from the starting point?

Show solution

North-East leg: 8 km East and 8 km North.
Then 2 km West: East part is $8 - 2 = 6$ km.
Net moves 6 and 8, so the distance is 10 km.

Answer

10 km

Type 4common2 practice Q

Total distance walked

How to spot it:

The question says "how much distance did he cover or walk" for a path that ends away from the start.

Method

Ignore all directions.
Add the length of every leg.
Do not use Pythagoras.

Why it works:

Distance walked counts every step. The straight line counts only the gap between start and end.

Try this

A man walks 9 km East and then 12 km North. What total distance does he cover?

Show solution

Add the two legs.
$9 + 12 = 21$ km.

Answer

21 km

Type 5very common

Left and right turns, then the gap

How to spot it:

The walker turns left or right in the middle of the walk. The straight-line distance is asked.

Method

Note the way he faces at the start.
After each turn, use the left-right table to find the new direction.
Turn each leg into a North, South, East or West move.
Find the net moves and the straight line.

Why it works:

The map never turns, so each new direction must be found one turn at a time.

Try this

Sanjay walks 8 km North. He turns right and walks 6 km. How far is he from the start?

Show solution

Facing North, right is East.
Moves: 8 km North and 6 km East.
6 and 8 give the triple 6-8-10.

Answer

10 km

Type 6occasional

How much shorter is the straight path

How to spot it:

The question asks how much distance would be saved by walking straight instead.

\text{saved} = \text{total walked} - \text{straight line}

Method

Add all the legs to get the total walked.
Find the straight line by Pythagoras.
Subtract the straight line from the total.

Why it works:

The saving is the gap between the two numbers you already know how to find.

Try this

A man walks 9 km East and then 12 km North. How much shorter is the direct path from start to finish?

Show solution

Total walked: 9 + 12 = 21 km.
Straight line: 15 km.
Saved: $21 - 15 = 6$ km.

Answer

6 km

Formula sheet

Shortest distance

d = \sqrt{x^2 + y^2}

x = net East-West move, y = net North-South move, after cancelling.

Number triples

3\text{-}4\text{-}5,\ 5\text{-}12\text{-}13,\ 8\text{-}15\text{-}17,\ 7\text{-}24\text{-}25

Doubles and triples of these also work: 6-8-10, 9-12-15.

Diagonal leg

k\sqrt{2}\ \text{along NE} = k\ \text{North} + k\ \text{East}

The same rule holds for NW, SE and SW.

Shortcuts that save time

⚡ Two-line tally

Keep one running total for East-West and one for North-South. Write West and South as minus. The two totals are the whole answer, with no drawing needed.

Example

A man walks 8 km East, 6 km West and then 3 km East. How far is he from the start, and in which direction?

Show solution

East-West: $8 - 6 + 3 = 5$ .
North-South: 0.

Answer

5 km, East

⚡ Spot the triple

If the two net moves are 3 and 4, or 5 and 12, or 8 and 15, or 7 and 24 (or their multiples), the answer is the third number. No square root is needed.

Example

A woman walks 24 km North and then 7 km East. How far is she from the start?

Show solution

24 and 7 are the short sides of the 7-24-25 triple.

Answer

25 km

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding all the legs when the question asks how far he is from the start.

Cancel the legs, then use the straight-line formula. Add all the legs only for 'distance walked'.

Mistake 02

Treating 'left' as West every time.

Find which way he faces first. Facing South, left is East.

Mistake 03

Turning the page when the walker turns.

Keep North at the top all the time. Only the walker turns.

Mistake 04

Splitting a $k\sqrt{2}$ diagonal leg into $k\sqrt{2}$ on each side.

A $k\sqrt{2}$ diagonal leg is $k$ on each side.

Mistake 05

Forgetting to cancel opposite legs.

Pair North with South and East with West before any square root.

Quick revision

Read this the night before the exam.

North is up and East is right. The page never turns.
Left and right depend on the way the walker faces.
Tally East-West and North-South. West and South are minus.
Cancel opposite legs before any calculation.
Straight line: $\sqrt{(\text{East})^2 + (\text{North})^2}$ .
Learn the triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
A diagonal leg of $k\sqrt{2}$ is $k$ on each of its two sides.

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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