Section Formula

Coordinate Geometry of Class 10

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 Section Formula

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 ALSection FormulaOX, BMSection FormulaOX, PNSection FormulaOX. 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.

Section Formula

Clearly, ΔAHP and ΔPKB are similar.

Section Formula

Section Formula

Now, Section Formula

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

Section Formula

And Section Formula

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

Section Formula

Thus the coordinates of P are Section Formula

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

Section Formula.

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