🧭 Chapter 07: Distance & Direction
Complete Learning Tutorial with Examples and Practice Questions
📍 1. Basic Directions
What is this?
Understanding the four cardinal directions and their relationships to solve movement-based problems.
🎯 The Four Cardinal Directions
| Direction |
Symbol |
Opposite |
Right Turn (90°) |
Left Turn (90°) |
| North ⬆️ |
N |
South |
East |
West |
| South ⬇️ |
S |
North |
West |
East |
| East ➡️ |
E |
West |
South |
North |
| West ⬅️ |
W |
East |
North |
South |
🧭 Direction Compass Visual
NORTH (N)
⬆️
|
|
WEST (W) ⬅️--+--➡️ EAST (E)
|
|
⬇️
SOUTH (S)
📐 Intermediate Directions (Diagonal)
| Direction |
Symbol |
Between |
Angle from North |
| North-East |
NE |
North & East |
45° |
| South-East |
SE |
South & East |
135° |
| South-West |
SW |
South & West |
225° |
| North-West |
NW |
North & West |
315° |
🎯 Pro Tip
Remember NEVER: North, East, West (going clockwise from top). Or use "NEWS" - North, East, West, South!
🔄 2. Turning Directions
What is this?
Understanding how turns affect your facing direction.
Type 1: Right Turn (Clockwise) 🔄
| Starting Direction |
After 90° Right |
After 180° Right |
After 270° Right |
| North (N) |
East (E) |
South (S) |
West (W) |
| East (E) |
South (S) |
West (W) |
North (N) |
| South (S) |
West (W) |
North (N) |
East (E) |
| West (W) |
North (N) |
East (E) |
South (S) |
Type 2: Left Turn (Anti-clockwise) 🔃
| Starting Direction |
After 90° Left |
After 180° Left |
After 270° Left |
| North (N) |
West (W) |
South (S) |
East (E) |
| East (E) |
North (N) |
West (W) |
South (S) |
| South (S) |
East (E) |
North (N) |
West (W) |
| West (W) |
South (S) |
East (E) |
North (N) |
📊 Quick Turn Reference
| Turn Type |
Angle |
Result |
| Right Turn |
90° |
Next direction clockwise |
| Left Turn |
90° |
Next direction anti-clockwise |
| About Turn / U-Turn |
180° |
Opposite direction |
| Three-quarter Right |
270° |
Same as 90° left |
| Three-quarter Left |
270° |
Same as 90° right |
💡 Example with Solution
| Step |
Action |
Direction |
Explanation |
| Start |
Facing |
North |
Initial position |
| Step 1 |
Turn Right 90° |
East |
N → E (clockwise) |
| Step 2 |
Turn Right 90° |
South |
E → S (clockwise) |
| Step 3 |
Turn Left 90° |
East |
S → E (anti-clockwise) |
| Final |
Now Facing |
East ✅ |
Final direction |
🎯 Pro Tip
For multiple turns, track each turn step-by-step. A 180° turn means you're facing the opposite direction!
📏 3. Distance Calculation
What is this?
Calculating the shortest distance between starting and ending points using Pythagoras theorem.
📐 Pythagoras Theorem
| Formula |
When to Use |
| Distance = √(x² + y²) |
When movement forms a right angle |
| x = Horizontal distance |
East-West movement |
| y = Vertical distance |
North-South movement |
🎯 Movement Direction Values
| Direction |
Horizontal (x) |
Vertical (y) |
| North |
0 |
+y |
| South |
0 |
-y |
| East |
+x |
0 |
| West |
-x |
0 |
💡 Complete Example
Problem: A person walks 3 km North, then 4 km East. What is the shortest distance from starting point?
| Step |
Direction |
Distance |
Position Change |
| Start |
- |
0 km |
Origin (0, 0) |
| Step 1 |
North |
3 km |
Position (0, 3) |
| Step 2 |
East |
4 km |
Position (4, 3) |
Calculation:
| Component |
Value |
Calculation |
| Horizontal (x) |
4 km |
East movement |
| Vertical (y) |
3 km |
North movement |
| Distance |
5 km |
√(4² + 3²) = √(16 + 9) = √25 = 5 km |
Answer: 5 km ✅
📊 Common Pythagorean Triplets
| x (km) |
y (km) |
Distance (km) |
Pattern |
| 3 |
4 |
5 |
3-4-5 |
| 5 |
12 |
13 |
5-12-13 |
| 6 |
8 |
10 |
6-8-10 |
| 8 |
15 |
17 |
8-15-17 |
| 9 |
12 |
15 |
9-12-15 |
| 12 |
16 |
20 |
12-16-20 |
🎯 Pro Tip
Memorize common Pythagorean triplets (3-4-5, 5-12-13, 8-15-17) to solve faster without calculations!
🗺️ 4. Position Finding
What is this?
Determining the final position or direction from the starting point.
Type 1: Final Direction from Start 🎯
| Example |
Problem |
Solution |
Answer |
| Example 1 |
Walk 5 km North, then 5 km East. In which direction from start? |
North (+5), East (+5) = North-East |
North-East |
| Example 2 |
Walk 3 km South, then 4 km West. In which direction from start? |
South (-3), West (-4) = South-West |
South-West |
Type 2: Distance and Direction Combined 📍
Complete Example:
Problem: Ramesh walks 10 km North, 6 km East, 10 km South, 2 km West. Find:
- Final distance from start
- Direction from start
| Step |
Direction |
Distance |
Net Position (x, y) |
| Start |
- |
- |
(0, 0) |
| Step 1 |
North |
10 km |
(0, +10) |
| Step 2 |
East |
6 km |
(+6, +10) |
| Step 3 |
South |
10 km |
(+6, 0) |
| Step 4 |
West |
2 km |
(+4, 0) |
Final Calculation:
| Component |
Value |
Explanation |
| Net Horizontal (x) |
+4 km |
6 km East - 2 km West = 4 km East |
| Net Vertical (y) |
0 km |
10 km North - 10 km South = 0 km |
| Distance |
4 km |
√(4² + 0²) = 4 km |
| Direction |
East |
Only horizontal movement remains |
Answer: 4 km East ✅
🎯 Pro Tip
Always calculate NET displacement: North cancels South, East cancels West!
🧮 5. Shadow-Based Direction
What is this?
Finding direction based on shadow position at different times of day.
☀️ Sun Position Throughout the Day
| Time |
Sun Direction |
Shadow Falls Towards |
| Morning (Sunrise) |
East |
West |
| Noon (12 PM) |
Overhead (North in Northern Hemisphere) |
Very Short |
| Evening (Sunset) |
West |
East |
📊 Shadow Direction Rules
| Sun Position |
Object Position |
Shadow Direction |
Example |
| Sun in East |
Person facing North |
Shadow towards West |
Morning time |
| Sun in West |
Person facing North |
Shadow towards East |
Evening time |
| Sun Overhead |
Any position |
Minimum shadow |
Noon time |
💡 Example Problems
| Problem |
Given |
Solution |
Answer |
| Example 1 |
Morning time, shadow falls to the left of a person |
Sun in East, shadow in West, person faces North |
North |
| Example 2 |
Evening time, Raj's shadow falls to his right |
Sun in West, shadow in East (right side), Raj faces South |
South |
| Example 3 |
Shadow points North-West |
Sun must be in South-East |
South-East |
🎯 Pro Tip
Shadow is ALWAYS opposite to the sun's direction! If shadow is in West, sun is in East.
🏃 6. Meeting Point Problems
What is this?
Two people starting from different points and finding where/when they meet.
Type 1: Opposite Direction Meeting 👥
Example: A and B are 100 km apart. A walks towards B at 30 km/hr, B walks towards A at 20 km/hr. When will they meet?
| Component |
Value |
Formula |
| Total Distance |
100 km |
Given |
| A's Speed |
30 km/hr |
Given |
| B's Speed |
20 km/hr |
Given |
| Combined Speed |
50 km/hr |
30 + 20 (moving towards each other) |
| Time to Meet |
2 hours |
Distance ÷ Combined Speed = 100 ÷ 50 |
Type 2: Right Angle Meeting 📐
Example: A walks 40 m North, B walks 30 m East from same point. What is the distance between them?
| Person |
Direction |
Distance |
Position |
| A |
North |
40 m |
(0, 40) |
| B |
East |
30 m |
(30, 0) |
| Calculation |
Value |
| Horizontal Gap (x) |
30 m |
| Vertical Gap (y) |
40 m |
| Distance Between |
50 m (√(30² + 40²) = √(900 + 1600) = √2500 = 50 m) |
🎯 Pro Tip
For opposite directions: ADD speeds. For same direction: SUBTRACT speeds!
🎓 7. Problem-Solving Strategy
6-Step Approach for Distance & Direction 📝
| Step |
Action |
Details |
| Step 1 |
Draw a Diagram |
Always sketch the path with directions marked |
| Step 2 |
Mark Starting Point |
Label it as O or Start |
| Step 3 |
Plot Each Movement |
Draw arrows for each movement with distance |
| Step 4 |
Calculate Net Displacement |
Find net horizontal (x) and vertical (y) |
| Step 5 |
Apply Formula |
Use Pythagoras: √(x² + y²) for distance |
| Step 6 |
Determine Direction |
Check final position relative to start |
📝 Complete Solved Example
Problem: A man walks 5 km East, then turns right and walks 12 km, then turns right again and walks 5 km. How far is he from the starting point and in which direction?
Step-by-Step Solution:
| Step |
Direction |
Distance |
Position (x, y) |
Diagram Notes |
| Start |
- |
- |
(0, 0) |
Origin point O |
| Step 1 |
East |
5 km |
(5, 0) |
Move right → |
| Step 2 |
Turn Right = South |
12 km |
(5, -12) |
Move down ↓ |
| Step 3 |
Turn Right = West |
5 km |
(0, -12) |
Move left ← |
Calculations:
| Component |
Value |
Working |
| Net Horizontal (x) |
0 km |
5 km East - 5 km West = 0 |
| Net Vertical (y) |
-12 km |
12 km South = -12 |
| Distance |
12 km |
√(0² + 12²) = √144 = 12 km |
| Direction |
South |
Only vertical displacement downward |
Answer: 12 km South from starting point ✅
📝 Practice Questions
Set A: Basic Direction Questions 🎯
| Q# |
Question |
Options |
Answer |
Explanation |
| Q1 |
If you are facing North and turn 90° right, which direction are you facing? |
A) SouthbrB) EastbrC) WestbrD) North |
B) East |
North → Right turn → East (clockwise) |
| Q2 |
A person walks 3 km North, then 4 km East. What is the shortest distance from start? |
A) 7 kmbrB) 5 kmbrC) 6 kmbrD) 4 km |
B) 5 km |
√(3² + 4²) = √25 = 5 km (Pythagorean triplet 3-4-5) |
| Q3 |
What is the opposite direction of North-East? |
A) North-WestbrB) South-EastbrC) South-WestbrD) West |
C) South-West |
Opposite of NE is SW (diagonal opposite) |
Set B: Intermediate Questions 📊
| Q# |
Question |
Options |
Answer |
Explanation |
| Q4 |
A man walks 10 km South, then 10 km East, then 10 km North. How far is he from start? |
A) 30 kmbrB) 10 kmbrC) 20 kmbrD) 0 km |
B) 10 km |
Net: 0 km vertical (10N-10S), 10 km East = 10 km East |
| Q5 |
In morning, sun is in East. If shadow falls to your right, which direction are you facing? |
A) NorthbrB) SouthbrC) EastbrD) West |
A) North |
Shadow in West (right side), means facing North |
| Q6 |
Ram walks 8 km North, then 15 km East. What is distance from start? |
A) 23 kmbrB) 17 kmbrC) 15 kmbrD) 20 km |
B) 17 km |
√(8² + 15²) = √(64 + 225) = √289 = 17 km |
Set C: Advanced Questions 🔥
| Q# |
Question |
Options |
Answer |
Explanation |
| Q7 |
A person walks 6 km East, 8 km North, 6 km West. Final position from start? |
A) 8 km NorthbrB) 8 km SouthbrC) 6 km EastbrD) 12 km North |
A) 8 km North |
Net: 0 horizontal (6E-6W=0), 8 km North |
| Q8 |
If you face East and make three right turns, which direction are you facing? |
A) EastbrB) WestbrC) NorthbrD) South |
D) South |
E→S (1st)→W (2nd)→N (3rd)...wait: E→S→W→N, so South! |
| Q9 |
Two people start from same point. A goes 12 km North, B goes 5 km South. Distance between them? |
A) 7 kmbrB) 17 kmbrC) 12 kmbrD) 13 km |
B) 17 km |
Total separation = 12 + 5 = 17 km (opposite directions) |
| Q10 |
A man walks 5 km towards East, turns left and walks 5 km, turns left again and walks 5 km. In which direction from start? |
A) NorthbrB) EastbrC) North-WestbrD) South-West |
C) North-West |
Path: 5E→5N→5W = Net position (-0+5E-5W, 0+5N) forms NW |
🎯 Key Takeaways
| # |
Key Concept |
Remember This |
| 1 |
Basic Directions |
NEVER = North, East, West (clockwise), South |
| 2 |
Right Turn |
Clockwise: N→E→S→W→N |
| 3 |
Left Turn |
Anti-clockwise: N→W→S→E→N |
| 4 |
Distance Formula |
√(x² + y²) for perpendicular movements |
| 5 |
Pythagorean Triplets |
3-4-5, 5-12-13, 8-15-17 (memorize!) |
| 6 |
Net Displacement |
North cancels South, East cancels West |
| 7 |
Shadow Rule |
Shadow is ALWAYS opposite to Sun |
| 8 |
Diagram First |
Always draw before calculating |
📊 Quick Reference: Direction Conversion
Turn Table (Starting from North)
| # of 90° Right Turns |
Direction |
# of 90° Left Turns |
Direction |
| 0 |
North |
0 |
North |
| 1 |
East |
1 |
West |
| 2 |
South |
2 |
South |
| 3 |
West |
3 |
East |
| 4 |
North |
4 |
North |
🏆 Common Mistakes to Avoid
| Mistake |
Why It's Wrong |
Correct Approach |
| Not drawing diagram |
Leads to confusion in complex problems |
Always sketch the path first |
| Forgetting to cancel |
Adding all distances instead of net |
North cancels South, East cancels West |
| Wrong turn direction |
Confusing left and right |
Use clockwise (right) / anti-clockwise (left) |
| Not using Pythagoras |
Direct addition of distances |
Use √(x² + y²) for perpendicular paths |
| Ignoring direction at end |
Finding distance but not direction |
Always check final position quadrant |
💡 Pro Tips for Exam Success
| Tip # |
Strategy |
Benefit |
| 1 |
Memorize Pythagorean triplets |
Saves 30-40 seconds per question |
| 2 |
Draw quick diagrams |
Prevents directional errors |
| 3 |
Mark N-E-S-W on every diagram |
Visual reference for turns |
| 4 |
Practice shadow problems |
Easy marks if you know the rule |
| 5 |
Double-check net displacement |
Most common mistake area |
