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Introduction
The Distance Between Two Points pattern teaches how to compute the straight-line distance (displacement) between two positions given their north-south and east-west movements or coordinates. This is fundamental in aptitude tests because many route and displacement problems reduce to finding the shortest (straight-line) distance.
Mastering this pattern helps solve map, navigation, and route-tracing questions quickly using coordinate thinking and the Pythagorean theorem.
Pattern: Distance Between Two Points
Key concept: Treat movements as coordinate changes on perpendicular axes. The straight-line distance = √(Δx² + Δy²), where Δx and Δy are the net east-west and north-south displacements respectively.
Steps to use:
- Convert movements into signed coordinates (East/ North positive, West/ South negative - or choose any consistent sign convention).
- Sum components separately to get net Δx (east-west) and Δy (north-south).
- Apply Pythagoras: distance = √(Δx² + Δy²).
- Report both magnitude and direction (quadrant) using the signs of Δx and Δy.
Step-by-Step Example
Question
A person starts at point O, walks 6 km North, then 8 km East. What is his straight-line distance from O and in which direction?
Solution
-
Step 1: Identify coordinate changes
Taking O as (0,0): move 6 km North → Δy = +6; then 8 km East → Δx = +8. -
Step 2: Compute net displacements
Net Δx = +8 (East), Net Δy = +6 (North). -
Step 3: Apply Pythagoras theorem
Distance = √(Δx² + Δy²) = √(8² + 6²) = √(64 + 36) = √100 = 10 km. -
Step 4: Determine direction
Δx > 0 and Δy > 0 → first quadrant → direction = North-East (specifically arctan(6/8) = 36.87° north of east). -
Final Answer:
10 km North-East -
Quick Check:
Recognize 6-8-10 as a Pythagorean triple (3-4-5 scaled by 2) → distance = 10 km ✅
Quick Variations
1. Movements include returns (e.g., North then South) - cancel opposite components first.
2. Use coordinates directly: given points (x₁,y₁) and (x₂,y₂), compute √((x₂-x₁)² + (y₂-y₁)²).
3. Mixed units or directions (km/m) - convert to same units before computing.
4. Angle asked: compute direction using tan⁻¹(|Δy/Δx|) and assign quadrant from signs.
Trick to Always Use
- Step 1: Always write net Δx and Δy with signs (East = +, West = -).
- Step 2: Cancel opposite movements before applying the formula.
- Step 3: Look for known Pythagorean triples (3-4-5, 5-12-13, etc.) to speed computation.
- Step 4: For direction, use θ = arctan(|Δy/Δx|) and adjust quadrant using sign of Δx, Δy.
Summary
- Treat east-west and north-south movements as perpendicular components.
- Always compute net displacement using √(Δx² + Δy²).
- Cancel opposite directions before applying the distance formula.
- Use direction ratios or tangent to specify angle when required.
Example to remember:
6 km North and 8 km East → Distance = 10 km North-East.