Section Formula.

 

Formula to find the coordinates of a point using section formula concept is as follows:

The line Segment joining points A(x1, y1) and B(x2, y2), is divided by the point P which has the coordinates (x, y).

  • The coordinates of the point dividing the line segment joining (x1, y1) and (x2, y2in the ratio m1 : m2 internally is given by ( (m1x2+m2x1)/(m1+m2),(m1y2+m2y1)/(m1+m2) )
  • The coordinates of the point dividing the line segment joining (x1, y1) and (x2, y2in the ratio m1 : m2 externally is given by ( (m1x2 - m2x1)/(m1 -m2),(m1y2  - m2y1)/(m1 - m2) )

Example 1:

Find the coordinates of the point which divides the line segment joining the points (4, – 3) and (8, 5) in the ratio 3 : 1 internally

Solution: Using section formula P(x, y) = { (m1x2 + m2x1)/(m1 + m2 ),(m1y2 + m2y1)/(m1 + m2 ) }

We get P(x, y) = { (3 X 8 + 1 X 4 )/(3+1),(3 X 5  + 1 X -3)/(3+1) }   = (7,3)

Example 2:

Find the coordinates of the point which divides AB, where A and B have the coordinates (2,4) and (4,6) respectively, in the ratio 1:3 (i) internally (ii) externally

Solution:

 The figure illustrates the two cases. Notice that in both cases AC: CB=1:3

                                        

We just have to put in the values in the section formula mentioned above

1. x= ((1×4)+(3×2)) / (1+3)= 5 / 2 and y=( (1×6)+(3×4) )/ (1+3)= 9 / 2 .

(a) Therefore the coordinates of the required point are (5/2,9/2)

2. (ii) x=((1×4)−(3×2) )/ (1−3)=1 and y= ( (1×6)−(3×4) ) / (1−3) =3

Therefore the coordinates of the required point are (1,3)

We need to ensure that we take care of the order of A and B. It is suggested to construct the points and  figure to avoid any mistakes in misinterpreting and avoid arriving at erroneous calculations.