2.3.3
Intuition Core idea kya haiEk line segment ka midpoint woh point hota hai jo dono ends ke bilkul beech mein hota hai.
Beech mein matlab: horizontal distance bhi equally split ho, AUR vertical distance bhi equally split ho.
Toh hum sirf x-coordinates ka average aur y-coordinates ka average lete hain.
Averaging = "beech mein milna".
Do points A ( x 1 , y 1 ) B ( x 2 , y 2 ) midpoint M A B A M = M B M = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
Hum formula memorise nahi karte — hum khud banate hain.
Intuition Feynman pictureA B x 2 − x 1 x 1
x-coordinate derive karna:
Dono points ke beech horizontal gap hai x 2 − x 1
Yeh step kyun? x-axis ke along distance sirf x-values ka difference hota hai.
Us gap ka aadha hai 2 x 2 − x 1
Kyun? "Midpoint" ka matlab literally half horizontal distance hota hai.
x 1 x M = x 1 + 2 x 2 − x 1
x 1 A
Simplify karo:
x M = 2 2 x 1 + x 2 − x 1 = 2 x 1 + x 2
Kyun? Common denominator 2 x 1 2 x 1 − x 1 = x 1
Bilkul same reasoning vertical direction mein:
y M = 2 y 1 + y 2
Bas yahi poora formula hai — sirf do averages hain. ✅
Worked example Compute karne se pehle predict karoPoints A ( 2 , 3 ) B ( 6 , 3 ) Forecast: y y 3 x 2 6 4 ( 4 , 3 ) Verify: x M = 2 2 + 6 = 4 y M = 2 3 + 3 = 3 ( 4 , 3 )
Worked example Example 1 — basic midpointA ( − 4 , 1 ) B ( 2 , 7 ) x M = 2 − 4 + 2 = 2 − 2 = − 1 Kyun? x's ka average lo; − 4 + 2 = − 2 y M = 2 1 + 7 = 2 8 = 4 Midpoint = ( − 1 , 4 )
Worked example Example 2 — missing endpoint find karnaM ( 3 , − 2 ) A ( 1 , 4 ) B ( x , y ) B 2 1 + x = 3 ⇒ 1 + x = 6 ⇒ x = 5 Pehle 2 se multiply kyun karein? Average ke andar ke "÷2" ko undo karne ke liye, taaki x 2 4 + y = − 2 ⇒ 4 + y = − 4 ⇒ y = − 8 B = ( 5 , − 8 )
Worked example Example 3 — midpoint se property prove karnaKya quadrilateral ke diagonals jo vertices P ( 1 , 1 ) , Q ( 5 , 3 ) , R ( 6 , 7 ) , S ( 2 , 5 ) P R ( 2 1 + 6 , 2 1 + 7 ) = ( 3.5 , 4 ) QS ( 2 5 + 2 , 2 3 + 5 ) = ( 3.5 , 4 ) Yeh important kyun hai? Same midpoint ⇒ diagonals apne middles par cross karte hain ⇒ figure ek parallelogram hai.
Haan, woh ek doosre ko bisect karte hain. ✔
Common mistake Add karne ki jagah subtract karnaGalat idea: "Midpoint mein 2 x 2 − x 1 sahi lagta hai kyunki distance mein subtraction use hoti hai.
Yeh fail kyun hota hai: 2 x 2 − x 1 half length hai, position nahi. Tum x 1 addition (averaging) chahiye.
Fix: Midpoint = average = sum ÷ 2 . Distance = difference. Dono ke kaam alag hain.
Common mistake Coordinates mix karnaGalat: x 1 y 2 2 x 1 + y 2 Tempting kyun lagta hai: Jab numbers paas paas hote hain tab careless ho jaate hain.
Fix: Kabhi axes mix mat karo. x's sirf x's ke saath average hote hain; y's sirf y's ke saath.
Common mistake Endpoint problems: double karna chahiye tha, divide kar diyaGalat: Missing endpoint find karne ke liye M Fix: 2 x 1 + x = x M x = 2 x M − x 1
Recall Feynman: ek 12-saal ke bachche ko explain karoTum aur tumhara dost ek number line par khade ho. Tumhare bilkul beech wala exact spot find karne ke liye, tum guess nahi karte — tum dono ki positions add karte ho aur aadha kar dete ho. Yeh karo kitna right/left ho (x) ke liye aur kitna up/down ho (y) ke liye. Woh do aadhe numbers hi "meeting point" hain. Wahi midpoint hai!
Mnemonic Yaad rakhne ka tarika"Add karo aur halve karo, x bhi y bhi — midpoint wahin milega jahan averages hain."
Yeh bhi: M idpoint = M ean (average).
Midpoint formula mein addition hoti hai ya subtraction? Kyun?
( 0 , 0 ) ( 8 , − 4 ) M ( 2 , 2 ) ( 0 , 0 )
What is the midpoint of A ( x 1 , y 1 ) B ( x 2 , y 2 ) ( 2 x 1 + x 2 , 2 y 1 + y 2 )
Hum coordinates ko subtract karne ki jagah ADD (average) kyun karte hain? Midpoint ek position hai; averaging woh point deta hai jo bilkul beech mein ho. Subtraction distance deta hai, location nahi.
Midpoint formula kaunse formula ka special case hai aur kis ratio ke saath? Section formula ka, ratio 1 : 1 m = n = 1
Midpoint M ( x M , y M ) A ( x 1 , y 1 ) B B = ( 2 x M − x 1 , 2 y M − y 1 )
( − 4 , 1 ) ( 2 , 7 ) ( − 1 , 4 )
Agar ek quadrilateral ke dono diagonals ka midpoint same ho, toh shape kya hai? Ek parallelogram (diagonals ek doosre ko bisect karte hain).
Section formula — midpoint 1 : 1 Distance formula — difference use karta hai; midpoint ke sum se contrast karo.Coordinate Geometry basics — points plot karna, axes.Properties of parallelograms — diagonals bisect each other.Centroid of a triangle — teen points ka average (averaging idea ko extend karta hai).
Midpoint M halfway between A and B
Walk x1 + half of gap x2-x1
Section formula ratio m:n
Diagonals bisect each other
