Direction & Distance

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

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Position of one point from another

⏱ 5 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

Some questions place two people, or a person and a fixed object, on the same map. Put the reference person at the middle of the map first. Then place everyone else, one sentence at a time. The distance and direction between any two people are then read straight off the picture.

Place the reference first

Two statements can link three people. "C is 5 km East of A. A is 2 km North of B. In which direction is C from B?"

The question asks about B. So put B at the middle of the map. Then hang the others on B.

B is at the middle.
A is 2 km North of B.
C is 5 km East of A, so C is 5 km East and 2 km North of B.

C is North-East of B.

Rule: Put the person named after "from" at the middle of the map. Everyone else hangs off that person.

Give every person a position

Put the common start at the middle of the map. Walk each person alone. Write each end point as (East, North).

Example: Amit ends at (3, 4). Sumit ends at (-4, -3). West and South are minus.

Distance between two people

The gap between two people is the straight line between their two end points.

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

The East gap is the difference of their East numbers. The North gap is the difference of their North numbers.

Example: Ravi walks 8 km North. Sam walks 15 km East. Ravi is at (0, 8) and Sam is at (15, 0). The gaps are 15 and 8, so the distance is 17 km.

Direction of one person from another

"Direction of B from A" means: stand at A, look at B. Find the East gap and the North gap from A to B. Read the compass word from their signs.

Watch: "A is East of B" and "B is East of A" give opposite pictures. Underline the person named after "of". That person is the reference.

Reverse the pair and the word flips. If C is North-East of B, then B is South-West of C.

Shared legs cancel

Suppose both people walk the same leg, such as "then both walk 20 km North". That leg moves both of them by the same amount. It does not change the gap between them.

Example: one walks 9 km East and the other walks 9 km West. Then both walk 20 km North. The gap stays 18 km.

Chains of positions

"P is 9 km West of Q. Q is 12 km South of R." Start at the last person, R, at the middle of the map. Q is 12 km South of R. P is 9 km West of Q. So P is 9 West and 12 South of R.

Distance from R to P: 9 and 12 give the triple, so 15 km. Direction of P from R: South-West.

Speed and time

Two people who walk at different speeds move different distances. Distance = speed times time. Then place both on the map.

Example: one walks North at 6 km per hour and the other walks East at 8 km per hour for 2 hours. They are 12 km North and 16 km East. The gap is 20 km.

Tip: Questions often ask "how far and in which direction". Find both. Options often pair the right distance with the wrong direction.

Question types you will see

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

Type 1very common3 practice Q

Distance between two walkers

How to spot it:

Two people start from the same place and walk in different directions. The gap between them is asked.

d = \sqrt{(\Delta E)^2 + (\Delta N)^2}

Method

Put the start at the middle of the map.
Find each person's end point as (East, North).
Find the East gap and the North gap.
Use the triples or the square root.

Why it works:

The gap is just the distance between two points on the map.

Try this

Two boys start from the same point. Ravi walks 8 km North and Sam walks 15 km East. How far apart are they?

Show solution

Ravi is at (0, 8) and Sam is at (15, 0).
The gaps are 15 East and 8 North.
15 and 8 give the triple 8-15-17.

Answer

17 km

Type 2very common4 practice Q

Direction of one person from another

How to spot it:

The question asks in which direction one person is from another after both have walked.

Method

Find both end points from the same start.
Take the reference person as the middle.
Find the East gap and the North gap from the reference to the other person.
Read the compass word from the two signs.

Why it works:

"B from A" is the arrow from A to B, and the signs of the gaps name it.

Try this

From the same point, Meera walks 6 km East and 10 km South. Nisha walks 2 km West and 7 km South. In which direction is Nisha from Meera?

Show solution

Meera is at (6, -10). Nisha is at (-2, -7).
From Meera to Nisha: 8 km West and 3 km North.
West and North make North-West.

Answer

North-West

Type 3common2 practice Q

Shared legs

How to spot it:

Both people walk the same leg, or one walks parallel to the other.

Method

Put the two people on the map after their own first legs.
A leg both walk in the same way does not change the gap.
Find the gap after the first legs.

Why it works:

The same leg moves both people equally, so it drops out when you subtract.

Try this

Two friends start together. One walks 9 km East and the other 9 km West. Then both walk 20 km North. How far apart are they?

Show solution

After the first legs the East gap is 9 + 9 = 18 km.
The 20 km North leg is shared, so the North gap stays 0.
The gap is 18 km.

Answer

18 km

Type 4occasional2 practice Q

Chain of positions

How to spot it:

Sentences such as 'A is East of B, B is North of C' link people. The pair not named together is asked.

Method

Put the last person of the chain at the middle.
Add each sentence's move one at a time.
Find the distance and direction of the asked pair.

Why it works:

Chained moves add up like the legs of a walk.

Try this

P is 9 km West of Q. Q is 12 km South of R. How far and in which direction is P from R?

Show solution

R is at the middle. Q is 12 km South of R.
P is 9 km West of Q, so P is 9 West and 12 South of R.
9 and 12 give 15. West and South make South-West.

Answer

15 km, South-West

Type 5common

Different speeds for the same time

How to spot it:

Two people move at given speeds for some hours. The gap between them is asked.

\text{distance} = \text{speed} \times \text{time}

Method

Find each person's distance = speed times time.
Put both on the map from the common start.
Find the gap from the two positions.

Why it works:

Distance from speed and time turns the question into a plain map problem.

Try this

From a crossing, Ajay cycles North at 6 km per hour and Bala cycles East at 8 km per hour. How far apart are they after 2 hours?

Show solution

Ajay: $6 \times 2 = 12$ km North.
Bala: $8 \times 2 = 16$ km East.
12 and 16 are 4 times 3 and 4 times 4, so the gap is 4 times 5.

Answer

20 km

Formula sheet

Distance between two points

d = \sqrt{(\Delta E)^2 + (\Delta N)^2}

Delta E is the East gap and Delta N is the North gap between the two people.

Distance from speed and time

\text{distance} = \text{speed} \times \text{time}

Use it first when two people walk at different speeds.

Shortcuts that save time

⚡ Start from the person named after 'from'

For "direction of C from B", draw B first at the middle. The chain then hangs off B. The answer is simply where C lands.

Example

A is 5 km North of B. C is 5 km East of A. In which direction is C from B?

Show solution

B at the middle. A is 5 km above B.
C is 5 km to the right of A. So C is 5 East and 5 North of B.

Answer

North-East

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Drawing A first when B is the reference person.

Put the person named after 'from' at the middle first.

Mistake 02

Giving 'C from B' when the question asked 'B from C'.

Underline the reference person in every sentence.

Mistake 03

Adding two distances that need Pythagoras.

Use the straight-line formula when the two moves are at right angles.

Mistake 04

Counting a leg both people walk as part of the gap.

A shared leg does not change the gap.

Mistake 05

Mixing up speed and distance.

Find distance = speed times time first, then place the person.

Quick revision

Read this the night before the exam.

Put the reference person at the middle of the map.
Write each person's end point as (East, North).
Gap: $\sqrt{(\Delta E)^2 + (\Delta N)^2}$ .
'B from A': stand at A, look at B. Reversing the pair flips the word.
A leg that both people walk does not change the gap.
Chains: start at the last person and add each move.
Speed times time gives distance.

Practice: 14 questions

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

Topic test · 10 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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