Section Formula

To find the co-ordinates of a point C (x, y) which divides the line segment joining the two points A (x₁,y₁) and B (x₂,y₂) in the ratio m:n ( internally and externally).

Proof:

Case (i)

C divides AB internally.

Let A (x₁, y₁) and B (x₂, y₂) be the two points joined by line segment AB. Let C (x, y) be the point on the line segment such that 

(In this case, AC and CB are real in the same direction on the line AB.)

Draw AP, CR and BQ perpendicular to x-axis.

AM perpendicular to CR and CM perpendicular to BQ.

AM = PR = x-x₁

CN = RQ = x₂-x

CM = y-y₁

BN = y₂-y

From the similar triangles, CAM and BCN, we have

Case (ii)

C divides AB externally.

From the similar triangles, CAM and CBN, we have

If i) r>0, C divides internally.

ii) r<0, C divides externally.

Note:

If C divides internally in the ratio 1:1 i.e., C is the midpoint of AB, then

This formula is called the midpoint formula.

Example:

Find the coordinates of the points A (-3, -4), B (-8,7) which divides the line segment joining the points A and B in the given ratio 5:7

i) internally and

ii) externally.

Suggested answer:

x₁ = -3, x₂ = -8

y₁ = -4, y₂ = 7

m = 5, n = 7

i) Internal division:

ii) External division: