Section Formula

Section Formulae

Simply, section formulae refer to the external and internal division of a line segment by a given point.
Section formulae have two types. They are,

1. Section formulae for an internal division.
2. Section formulae for an external division.
   
   

Section Formulae for Internal Division
Let's take a line with two ends point A(x₁, y₂) and B(x₂, y₂) which are joined by the line segment AB. Consider P(x, y) be any point on AB which divides the line internally in the ratio m₁:m₂
i.e. AP:PB = m₁:m₂
The formula for the section formulae in internal division is (x, y) = (\(\frac{m_1x_2+m_2x_1}{m_1+m_2}\)),(\(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\))

Section Formulae for External Division
If the point P(x, y) divides AB externally in the ratio of m₁:m₂ then the divided segment BP is measured in opposite direction and hence m₂ is taken as negative.

\(\therefore\) The section formulae for external division is,
(x, y) = (\(\frac{m_1x_2 - m_2x_1}{m_1 - m_2}\)),(\(\frac{m_1y_2 - m_2y_1}{m_1 - m_2}\))

In special case, the midpoint formulae is also used'
m₁:m₂ = 1:1 i.e. m₁ = m₂
\(\therefore\) x = \(\frac{x_1 + x_2}{2}\) and y = \(\frac{y_1 + y_2}{2}\)
Thus, co-ordinates P(x, y) are P(\(\frac{x_1 + x_2}{2}\),\(\frac{y_1 + y_2}{2}\)) which is called mid-point formulae.