SECTION FORMULA EXAMPLE PROBLEMS WITH SOLUTIONS

Section Formula Example Problems With Solutions :

We use the section formula to find the point which divides the line segment in a given ratio. 

The point P which divides the line segment joining the two points A (x₁,  y₁) and B (x₂, y₂) internally in the ratio l : m is  

If P divides a line segment AB joining the two points A (x₁,  y₁) and B (x₂, y₂) externally in the ratio l : m is,

Section Formula Example Problems

Example 1 :

Find the ratio in which the line segment joining the points (-3,10) and (6,-8) is divided by (-1,6).

Solution :

Here x₁  =  -3, y₁  =  10, x₂  =  6, y₂  =  -8 

[m(6) + n(-3)]/(m + n), [m(-8) + n(10)]/(m + n)  =  (-1, 6)

        (6m - 3n)/(m + n), (-8m + 10n)/(m + n) = (-1, 6)

Equating the coefficients of x and y,

(6m - 3n)/(m + n)  =  -1 ---(1)       (-8m + 10n)/(m + n)  =  6

From (1),

6m -  3n  =  -1 (m + n)

6m - 3n  =  -1 (m + n)

6m - 3n  =  -m - n

6m + m  =  -n + 3n

7m  =  2n

m/n  =  2/7

m : n  =  2 : 7

So, the point (-1,6) is dividing the line segment in the ratio 2 : 7.

Example 2 :

Find the ratio in which the line segment joining A (1, -5) and B(-4, 5) is divided by the x axis. Also find the coordinates of the point of division.

Solution :

Let m:n be the ratio which divides the line segment joining the points A(1, -5) and B(-4, 5)

x₁  =  1, y₁  =  -5, x₂  =  -4 , y₂  =  5 

[m(-4) + n(1)]/(m + n) , [m(5) + n(-5)]/(m + n)  =  (x, 0)

(-4m + n) /(m + n) , (5m - 5n)/(m + n)  =  (x,0)

(-4m + n)/(m + n)  =  x      (5m - 5n)/(m + n)  =  0

(5m - 5n)/(m + n)  =  0

5 m - 5 n  =  0

5 m = 5 n

m/n = 5/5  ==>  m : n = 1 : 1

Example 3 :

If (1, 2), (4, y) (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y

Solution :

Let A (1, 2), B (4, y) C (x, 6) and D (3, 5) are the vertices of a parallelogram.

In a parallelogram midpoint of the diagonals will be equal.

Midpoint of diagonal AC  =  Midpoint of diagonal BD

Midpoint  =  (x₁ + x₂)/2 , (y₁ + y₂)/2

Midpoint of diagonal AC :

x₁  =  1, y₁  =  2, x₂  =  x , y₂  =  6 

  =  (1 + x)/2, (2 + 6)/2

  =  (1 + x)/2, 8/2

  =  (1 + x)/2, 4 ------(1)

Midpoint of diagonal BD :

x₁  =  4, y₁  =  y, x₂  =  3 , y₂  =  5 

  =  (4 + 3)/2, (y + 5)/2

  =  7/2, (y + 5)/2 ------(2)

Equating x and y coordinates, we get

After having gone through the stuff given above, we hope that the students would have understood, section formula example problems with solutions.

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