Section Formula

Section Formula and Some Solved Examples

Section formula is a part of coordinate geometry which is used extensively in subjects like physics. Finding out the centroid of a triangle, excenters and incenters of a triangle, the center of the mass of systems, and so on can be done with the help of a section formula. Mathematically speaking, section formula is a formula of coordinate geometry which is used to obtain the ratio in which a line segment is segregated by a point internally or externally. In this article, we will discuss the section formula as well as the distance formula and solve some examples related to it.

Section Formula

Suppose, you decided to go to a fair from point X to point Y. Before reaching the desired destination, one of the tires of your bicycle punctures at point B. Now, you want your brother to help you who is at point X. How will you tell your brother the exact location where your tire has punctured? Use section formula. How? Let us understand. When a line segment XY is split by point B in the ratio p:q, the section formula is used to determine the coordinates of point B. The section formula is further divided into two types, namely, the internal section formula and the external section formula. Let us discuss them visit here

  • Internal Section Formula: Internal section formula is used when any given line segment is split internally in the ratio p:q. Another phrase that we use in place of the internal section formula is “section formula for interior division”. Given below is the internal section formula:

(px2 + qx1)/ (p+q) for the x coordinate of point B and (py2 + qy1)/ (p+q) for the y coordinate of point B.

  • External Section Formula: External section formula is used when point B lies on the exterior part of the given line segment. Another phrase that we use in place of the external section formula is “section formula for exterior division”. Given below is the external section formula:

           (px2 – qx1)/ (p-q) for the x coordinate of point B and (py2 – qy1)/ (p-q) for the y

           coordinate of point B.

Some Solved Examples

  1. Find the midpoint of the line segment XY where the coordinates of X are (6, 8) and the coordinates of Y are (16, 24).

      Answer: It is given that x1= 6, x2= 16, y1= 8 and y2= 24.

     The midpoint formula is (x1 + x2) /2 for the x coordinate and (y1 + y2) /2 for the y

     coordinate.

     By substituting the given values, we get the coordinates as (6+16) /2, (8+24) /2 =

     (11,16). Thus the coordinates of the midpoint of the line segment XY are (11,16).

  1. What are the coordinates of point M, which splits the line segment joining the point (6,3) and (9,12) in the ratio 3:1 internally?

     Answer: Let the required coordinates of the point M be (x,y).

     According to the internal section formula, we know that: (px2 + qx1)/ (p+q) for the x

     coordinate of point B and (py2 + qy1)/ (p+q) for the y coordinate of point B.

     Thus, x= (3* 9 + 1 * 6)/ 3+1 = 8.25 and y= (3* 12 + 1* 3)/ 4 = 9.75.

Midpoint Formula

The midpoint formula comes in handy when we need to find the coordinates of a point that lies completely at the middle of any given line segment. It is a subset of the section formula which means that the midpoint formula is obtained with the help of the section formula itself. Let us now learn about the midpoint formula. The midpoint formula is given below:

(x1 + x2) /2 for the x coordinate and (y1 + y2) /2 for the y coordinate.

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