Section Formula

FORMULA FOR INTERNAL DIVISION:

The coordinates of the pint which divided the line segment joining the pints (x₁, y₁) and x₂, y₂) internally in the ratio m : n

are given by

Proof: Let O be the origin and let OX and OY be the X-axis and Y-axis respectively. Let A(x₁, y₁) and B(x₂, y₂) bet the given points. Let (x, y) be the coordinates of the point p which divides AB internally in the ratio m : n Draw ALOX, BMOX, PNOX. Also draw AH and PK perpendicular from A and P on PN and BM respectively. Then

OL = x₁, ON = x, OM = x₂, AL = y₁, PN = y and BM = y₂.

∴ AH = LN = ON – OL = x – x₁, PH = PH – HN

= PN – AL = y – y₁, PK = NM = OM – ON = x₂ - x

and BK = BM – MK = BM – PN = y₂ – y.

Clearly, ΔAHP and ΔPKB are similar.

∴

⇒

Now,

⇒ mx₂ – mx = nx – nx₁ ⇒ mx + nx = mx₂ + nx₁

⇒

And

⇒ my₂ – my = ny – ny₁ ⇒ my + ny = my₂ + ny₁

⇒

Thus the coordinates of P are

FORMULA FOR EXTERNAL DIVISION:

The coordinates of the points which divides the line segment joining the points (x₁, y₁) and (x₂, y₂) externally in the ratio m : n are given by

.