Concept 7.2
The Section Formula
Source: NCERT Mathematics, Class 1012 minute lesson
Formal statement
The point P that divides the segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n is P((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
The intuition
If P divides AB in the ratio 1 : 3, it is a quarter of the way from A. So P sits close to A: A's coordinates get more weight.
That's why the weights cross over: m multiplies B's coordinate and n multiplies A's. Test it: with m = 0, P should be A, and only the crossed version gives that.
The attention
Look at which weight goes with which end: they cross over.
x = ( m·x₂ + n·x₁ ) / (m + n) not ( m·x₁ + n·x₂ ) / (m + n)
Look hereNot here
- Look here
- m with B, n with A. m sits next to x₂, the far end. P is m parts from A, so B's coordinate gets the weight m.
- Not here
- m with A, n with B. Pairing m with x₁ feels natural, because both are written first. It puts P on the wrong side of the middle.
- What it teaches
- Sanity-check a formula with an extreme case: if m = 0, P should be at A.