2.3.2
Agar aap do points ke beech straight-line distance find karna chahte ho, toh aap basically ye pooch rahe ho: "Agar main seedha udke wahan jata, toh kitna safar karta?" Iska jawab milta hai ek right triangle banane se jahan distance hypotenuse hoti hai. Horizontal aur vertical legs sirf coordinates ke differences hote hain, aur Pythagoras hume batata hai inhe kaise combine karna hai.
Definition Do Points Ke Beech DistancePoints P ( x 1 , y 1 ) Q ( x 2 , y 2 ) distance d d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
YE FORMULA KYU? Kyunki do points ke beech sabse chhota rasta ek straight line hai, aur woh line ek right triangle ki hypotenuse hai jiske legs grid lines ke saath chalte hain.
HUM KYA KAR RAHE HAIN: Coordinate differences se ek right triangle construct kar rahe hain, phir Pythagoras' theorem apply kar rahe hain.
Do points plot karo P ( x 1 , y 1 ) Q ( x 2 , y 2 )
P Q
P x Q y Ye dono point R ( x 2 , y 1 )
Legs ki lengths nikalo:
Horizontal leg P R y y 1 x x 1 x 2 P R = ∣ x 2 − x 1 ∣ ABSOLUTE VALUE KYU? Distance hamesha positive hoti hai, lekin x 2 − x 1
Vertical leg QR x x 2 y y 1 y 2 QR = ∣ y 2 − y 1 ∣
Right triangle P QR
P Q 2 = P R 2 + Q R 2 KYU? Kisi bhi right triangle mein, (hypotenuse)² = (leg₁)² + (leg₂)²
Leg lengths substitute karo:
d 2 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 AB ABSOLUTE VALUE SIGNS KYU NAHI? Kyunki squaring kisi bhi number ko positive bana deti hai: ( − a ) 2 = a 2
Square root lo:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 SIRF POSITIVE ROOT KYU? Distance negative nahi ho sakti.
Worked example Example 1: Simple Integer CoordinatesA ( 1 , 2 ) B ( 4 , 6 )
Solution: Identify karo: x 1 = 1 , y 1 = 2 , x 2 = 4 , y 2 = 6
d = ( 4 − 1 ) 2 + ( 6 − 2 ) 2 Ye step kyun? Formula mein substitute kar rahe hain.
= ( 3 ) 2 + ( 4 ) 2 Ye step kyun? Differences simplify kar rahe hain.
= 9 + 16 Ye step kyun? Har term ko square kar rahe hain.
= 25 = 5 Ye step kyun? Add karke square root le rahe hain.
Answer: Distance 5 units hai.
Physical meaning: Agar har unit 1 km hai, toh points straight-line distance mein 5 km door hain.
Worked example Example 2: Negative CoordinatesP ( − 3 , 2 ) Q ( 5 , − 4 )
Solution:
Identify karo: x 1 = − 3 , y 1 = 2 , x 2 = 5 , y 2 = − 4
d = ( 5 − ( − 3 ) ) 2 + ( − 4 − 2 ) 2 Ye step kyun? Direct substitution. Negative signs ke saath careful raho!
= ( 5 + 3 ) 2 + ( − 6 ) 2 Ye step kyun? Negative ko subtract karna matlab addition hai: 5 − ( − 3 ) = 5 + 3
= ( 8 ) 2 + ( − 6 ) 2 Ye step kyun? Simplify kar rahe hain.
= 64 + 36 Ye step kyun? Note karo ki ( − 6 ) 2 = 36 − 36
= 100 = 10 Answer: Distance 10 units hai.
Worked example Example 3: Irrational ResultM ( 0 , 0 ) N ( 3 , 4 )
Solution:
d = ( 3 − 0 ) 2 + ( 4 − 0 ) 2 = 9 + 16 = 25 = 5 Ab try karo M ( 0 , 0 ) aur N ( 2 , 3 ) d = ( 2 ) 2 + ( 3 ) 2 = 4 + 9 = 13 13 ≈ 3.606
Insight: Sabhi distances whole numbers nahi hoti! Zyaadatar distances irrational hoti hain .
Common mistake Mistake 1: Differences ko Square Karna Bhool JanaGalat: d = ( x 2 − x 1 ) + ( y 2 − y 1 ) Kyun sahi lagta hai: Aap soch sakte ho "bas horizontal aur vertical distances add karo."
Fix: Woh tumhe Manhattan distance dega (grid lines ke saath chalna), straight-line distance nahi. Pythagoras ko legs ke squares chahiye:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Visual: 3 blocks east phir 4 blocks north chalna total 7 blocks hai (Manhattan), lekin straight-line distance 5 blocks hai (Pythagorean).
Common mistake Mistake 2: Negative Coordinates Ke Saath Sign ErrorsGalat: P ( − 2 , 3 ) Q ( 1 , − 1 ) ( 1 − 2 ) 2 ( 1 − ( − 2 ) ) 2
Kyun sahi lagta hai: Aap x 1 = − 2
Fix: Hamesha carefully substitute karo:
x 2 − x 1 = 1 − ( − 2 ) = 1 + 2 = 3 y 2 − y 1 = − 1 − 3 = − 4
Phir square karo: ( 3 ) 2 = 9 ( − 4 ) 2 = 16 d = 25 = 5
Common mistake Mistake 3: Har Term Ka Alag-Alag Square Root LenaGalat: ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 = ( x 2 − x 1 ) + ( y 2 − y 1 )
Kyun sahi lagta hai: Lagta hai jaise tum squares ko "undo" kar rahe ho.
Fix: a 2 + b 2 = a + b
Sahi: 9 + 16 = 25 = 5
Galat: 9 + 16 = 3 + 4 = 7
Distance formula in chezon ki foundation hai:
Analytical geometry (circles, ellipses, hyperbolas sab distance use karte hain)3D distance (sirf ( z 2 − z 1 ) 2 Optimization problems (closest points dhundhna)Physics (displacement magnitude, distance-time calculations)
Recall Ek 12-Saal-Ke Bachche Ko Explain KaroSocho tum ek rectangular maidan ke ek corner par ho, aur tumhara dost opposite corner par hai. Tum edges ke saath chal sakte ho (pehle right, phir upar), lekin woh lamba rasta hai. Agar tum seedha ghaas par chalo, toh kitni door jaoge?
Jawab rectangle ki hypotenuse hai. Use nikalne ke liye, tum measure karo kitna across (horizontal) aur kitna upar (vertical), phir Pythagoras use karo: dono ko square karo, add karo, square root lo. Yahi distance formula hai! Ye tab bhi kaam karta hai jab tumhara dost kisi ajeeb direction mein ho (jaise southwest) kyunki hum coordinate differences use karte hain jo automatically direction handle karte hain.
Mnemonic Formula Yaad Karo"Difference Squared, Sum Under Root"
D ifference in x y S quare karo dono differences koS um karo unheU nder the square root
Ya picture karo: "Pythagoras on the Plane" — koi bhi do points axes ke saath ek right triangle banate hain.
Pythagoras Theorem — is derivation ki foundationCartesian Coordinate System — wo framework jahan points rehte hainSection Formula — line segments divide karne ke liye distance use karta haiEquation of Circle — center se fixed distance par saare points ke roop mein define hota haiDistance Formula in3D — teen dimensions tak natural extensionMidpoint Formula — distance se related, center point dhundhta hai
#flashcards/maths
What is the distance formula for points ( x 1 , y 1 ) ( x 2 , y 2 ) d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Which theorem is used to derive the distance formula? Pythagoras' Theorem applied to a right triangle formed by coordinate differences
Why do we square the coordinate differences in the distance formula? To apply Pythagoras' theorem (hypotenuse² = leg₁² + leg₂²) and to eliminate the need for absolute values (since squaring makes any number positive)
What shape is formed when deriving the distance formula? A right triangle with horizontal leg ( x 2 − x 1 ) ( y 2 − y 1 )
Find the distance between ( 1 , 2 ) ( 4 , 6 ) d = ( 4 − 1 ) 2 + ( 6 − 2 ) 2 = 9 + 16 = 25 = 5
True or False: a 2 + b 2 = a + b
If x 1 = − 3 x 2 = 5 x 2 − x 1 5 − ( − 3 ) = 5 + 3 = 8
Why is the distance formula symmetric? Swapping the two points gives ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
What is the distance from origin ( 0 , 0 ) ( 3 , 4 ) d = 3 2 + 4 2 = 9 + 16 = 5
Why do we only take the positive square root in the distance formula? Distance is always non-negative; a negative distance has no physical meaning
squaring removes abs value
Square coordinate differences
Take positive square root
