Section formula — internal and external division

2.3.4 · Maths › Coordinate Geometry

WHAT — hum kya dhundh rahe hain?

Figure — Section formula — internal and external division

HOW — scratch se derive karte hain

Maano $A = (x_{1}, y_{1})$ , $B = (x_{2}, y_{2})$ , aur $P = (x, y)$ , $A B$ ko internally ratio $m : n$ mein divide karta hai.

Similar triangles kyun? $A$ , $P$ , $B$ se $x$ -axis par verticals girao. Horizontal gaps usi proportion mein badhte hain jis proportion mein points line ke saath move karte hain, kyunki line ki slope constant hoti hai. Isliye ratio $A P : P B$ , $x$ -shadows aur $y$ -shadows mein bilkul identically dikhta hai.

$x$ -axis par project karo. $A$ se $P$ tak jaane mein $x - x_{1}$ cover hota hai; $P$ se $B$ tak $x_{2} - x$ cover hota hai. Yeh ratio $m : n$ mein hain: $\frac{x - x _{1}}{x _{2} - x} = \frac{m}{n} .$ Kyun? Kyunki ek straight line horizontal distance ko exactly usi ratio mein split karti hai jis mein woh segment length ko split karti hai.

Cross-multiply karo: $n (x - x_{1}) = m (x_{2} - x) \Rightarrow n x - n x_{1} = m x_{2} - m x .$ $x$ collect karo: $m x + n x = m x_{2} + n x_{1} \Rightarrow x = \frac{m x _{2} + n x _{1}}{m + n} .$ $y$ ke liye bhi exactly yahi: $y = \frac{m y _{2} + n y _{1}}{m + n} .$

External case. Ab $P$ bahar hai, isliye $A \to P$ aur $P \to B$ opposite directions mein point karte hain. Projected lengths mein se ek negative ho jaati hai. $n$ ko $- n$ se replace karo: $P = (\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n}) .$

Midpoint = internal division with $m = n = 1$ : $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) .$

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bache ko explain karo

Socho ek rope hai ghar $A$ se ghar $B$ tak. Tum ek aisi jagah khade hona chahte ho jo "har 1 step ke liye 2 step" wali ho. Kahan ho yeh dhundne ke liye tumhe chalna nahi padta — bas dono gharon ke addresses ko mix karo: $B$ ka address $2$ parts lo aur $A$ ka address $1$ part lo, phir $3$ (total parts) se divide karo. Woh mixed address tumhari jagah hai. External tab hota hai jab tum ek ghar ke aage bhi chalte rehte ho instead of beech mein rehne ke — toh addresses add karne ki jagah tum unhe subtract karte ho.

Connections

Midpoint formula — special case $m = n = 1$ .
Centroid of a triangle — har median ko vertex se $2 : 1$ mein divide karta hai.
Similar triangles — derivation ka geometric engine.
Distance formula — ratios ko independently verify karne ke liye use hota hai.
Slope of a line — constant slope hi wajah hai ki shadow ratios match karte hain.
Collinearity of three points — teen points collinear hain tab hi jab koi ek dono ko kisi ratio mein divide kare.

Internal section formula for ratio

m : n

(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

External section formula for ratio

m : n

(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})

Which point's coordinates does

m

multiply?

Woh door wala point

B

ke coordinates (kyunki

\frac{A P}{P B} = \frac{m}{n}

Midpoint as a special case

m = n = 1 \Rightarrow (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

What does a negative ratio

m : n

signify?

External division.

Why does external division use

m - n

n

- n

se replace karo (projected lengths ki opposite direction).

Geometric principle behind the derivation

Similar triangles / axes par proportional projections.

Divide

A (2, 3), B (8, 9)

internally

2 : 1

(6, 7)

When does external division give no finite point?

Jab

m = n

ho (denominator

m - n = 0

Centroid uses which ratio on a median?

2 : 1

vertex se measure karke.

WHAT — hum kya dhundh rahe hain?

HOW — scratch se derive karte hain

Worked examples

Common mistakes

Connections

Concept Map