SECTION FORMULA EXAMPLE PROBLEMS WITH SOLUTIONS

We can 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,

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 

Equating the coefficients of x and y,

(6l - 3m)/(l + m) = -1 ----(1)

(-8l + 10m)/(l + m) = 6  ----(2)

6l -  3m  =  -1 (l + m)

6l - 3m  =  -l - m

6l + l = -m + 3m

7l = 2m

l/m  =  2/7

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

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

 (-4l + m)/(l + m)  =  x -----(1)

 (5l - 5m)/(l + m)  =  0-----(2)

(5l - 5m)/(l + m)  =  0

5l - 5m  =  0

5l = 5m

l/m = 5/5  ==>  l : m = 1 : 1

By applying the ration in (1), we get

x = (-4(1) + 1)/(1 + 1)

x = -3/2

So, the point of intersection is (-3/2, 0).

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  =  (x1x2)/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

(1 + x)/2, 4  =  7/2

1 + x  =  7

x  =  7 - 1

x  =  6

(y + 5)/2  =  4

4  =  (y + 5)/2

8  =  y + 5

y  =  8 - 5  = 3

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