Section Formula

The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio .

The midpoint of a line segment is the point that divides a line segment in two equal halves. The section formula builds on it and is a more powerful tool; it locates the point dividing the line segment in any desired ratio.

The section formula is helpful in coordinate geometry; for instance, it can be used to find out the centroid, incenter and excenters of a triangle. It has applications in physics too; it helps find the center of mass of systems, equilibrium points, and more.

Internal Divisions with Section Formula

If point  lies on line segment  between points  and  and satisfies  then we say that divides  internally in the ratio  The point of division has the coordinates

The formula can be derived by constructing two similar right triangles, as shown below. Their hypotenuses are along the line segment and are in the ratio .

The red and the green triangles are similar since the corresponding angles of the triangles are equal. This implies that the ratio of their corresponding sides are equal. Note that point  is  away from . That is,

Similarly, solving for  gives

Therefore, from  and 

As a special case of internal division, if  is the midpoint of , then it divides  internally in the ratio . Hence applying the formula for internal division and substituting , we get

Given  and , what are the coordinates of point which internally divides line segment  in the ratio 

The point  is  away from point .

When measured parallel to the -axis, we get

When measured parallel to the -axis, we get

Thus, the coordinates of  are  

Given , what are the coordinates of  if point  divides line segment internally in the ratio 

Solution:

In this example, we are to find one of the endpoints of the line segment. Drawing similar triangles will help us solve this problem too.

The sides of the triangle are in the ratio . The base of the pink triangle has length . The base of the green triangle is three times as long, that is, . Solving this yields .

The height of the pink triangle is . The height of the green triangle is three times as long, that is, . Solving this equation yields .

Thus, the coordinates of  are (1,-2)

Given  and , point  internally divides line segment  in the ratio . If  is the intersection point of  and the -axis, what is the value of 

Since point  is on the -axis, its coordinate is zero. We can write the coordinates of  as .

The horizontal distance between  and  is . 
The horizontal distance between  and  is .

The ratio of the bases of the right triangles is , or . Since the triangles are similar, the ratio of their hypotenuses is also .

Therefore, point  divides line segment  in the ratio . 

In what ratio does the point divide the line segment joining  and 

We can draw 2 similar right triangles: the red triangle with hypotenuse  and the blue triangle with hypotenuse 

Point  divides line segment  in the ratio , which is equivalent to  since the triangles are similar. Let us find the lengths of  and 

Thus, point  divides line segment in the ratio .

Alternatively, the ratio  is also equal to  i.e.

We get the ratio  again, which is consistent with our previous calculations. 

To solve questions similar to the above example there is an alternative method in which you need to solve only for one variable instead of two variables. Below given example demonstrates it.

Find the ratio in which the point divides the line joining points  and .

You can solve this question in normal method by assuming the ration to be . But now we will take a different substitution. Assume  so that . Now the required ratio will be .Given that the point . So, substitute either  or  in the above result.

.

Find the co-ordinates of point  which divides the line joining  and  in the ratio .

Let the co-ordinates of  be .

Find