Section Formulae

A line segment could be divided into two parts by placing a point on the line between the two extreme ends of the line segment or someplace outside of the line segment. Section formulas can also be used in coordinate geometry to determine a relationship between the coordinates of the point that divides a line segment into two parts and the ratio in which it is divided. For example, we can use the section formula in mathematics to calculate the proportion in which the line segment is divided by the point if we are given the coordinates of the point that divides the line segment. The section formulae are also used to find the coordinates of a point that divides a line segment connecting two locations into two portions with a length ratio of m:n.

Section Formula in Coordinate Geometry

When any point on the line segment divides into two parts, the formula used to calculate the coordinates of that specific point is called the section formula. For example, let us assume that we have a point A(x,y) which divides any line segment that is marked as points P (x1,y1) and Q(x2,y2). To determine the coordinates, we will use the section formula that is mathematically written as:

P(x,y) = [(mx2+nx1)/m+n], [(my2+ny1)/m+n]

The section formula is classified into two types:

• Internal division
• External division

Internal Section Formula 

The internal section formula is applied when a line segment is divided internally by a point in the desired ratio. This formula is used to find the coordinates of a point in the given balance between two points and on the line that connects them.

Then, the coordinates of the point P in terms of the given ratio and for the given coordinates of the points A(x1,y1), B(x2,y2) are:

P(x,y) = (mx2+nx1/m+n, my2+ny1/m+n)

External Section Formula

The external section formula is applied when a line segment is divided externally by a point in a specific ratio. In the given ratio, this formula is used to discover the coordinates of the point on the line segment linking the two points and falling beyond them.

P(x,y) = (mx2-nx1/m-n, my2-ny1/m-n)

Midpoint Formula for Sections

The midpoint formula determines the coordinates of a line segment’s midpoint. The ratio between the two sections is 1:1 in this case. A midpoint is a point in the line segment in the middle. The point M(x,y) is the midpoint of the line segment, as shown in the formula given below.

M(x,y) = (x2+x1/2,y2+y1/2)

Conclusion

In the above notes, we have studied the Section formulae. Section formula is used to determine the point coordinates that externally or internally divide a line segment into equal ratios. The point on a line segment divides that line into two parts which are supposed or not supposed to be equal. A ratio that the point divides the line segment could be found when we know the coordinates of that given point. It is also possible to determine the point of the division when we already know the ratio of the line segment joining the two given points. These two properties can be gained with the assistance of the section formula in the coordinate geometry. This crucial formula helps determine the incentre, excentric, and centroid in mathematics. In physics, it is used to calculate equilibrium points and the centre of mass.