An Idea on Section Formula

 Coordinate geometry is the study of geometric figures using coordinate axes. In the coordinate axes, straight lines, curves, circles, ellipses, hyperbolas, and polygons may all be easily drawn and resized.

To get the coordinates of the point that separates a line segment into a ratio, apply the Section formula (externally or internally). In physics and mathematics, this formula is often employed. In mathematics, it is used to discover the centroid, incenters, and excentres of a triangle, and in physics, it is used to find the center of mass, equilibrium points, and other things. The section formula is widely used to determine the midpoint of a line segment.

Section Formula: 

We use the section formula to discover the coordinates of a point that splits a line segment externally or internally in certain ratio. It’s a useful tool for determining the coordinates of a point by splitting a line segment in a certain ratio. This section formula may also be used to determine the midpoint of a line segment as well as to derive the midpoint formula.

Section Formula Definition in Coordinate Geometry:

The section formula is the method used to obtain the coordinates of a point on a line segment that splits it into two segments. Assume we have a point P(x,y) that splits a line segment labelled as A (x1,y1) and B (x1,y2) (x2,y2). We utilise the section formula to obtain the coordinates, which is mathematically stated as: P (x, y) = ( (mx2 + nx1)/ (m + n), (my2 + ny1) / ( m + n))

• Note that m and n are used to divide the line segment into a m:n ratio.

In coordinate geometry, the section formula is split into sub-formulas, which are:

• Internal Section Formula 

• External Section Formula 

Internal Section Formula 

When a line segment is divided internally by a point in a specific ratio, the internal section formula is applied. This formula is used to calculate the coordinates of a point in the specified ratio that lies between two points and on the line that connects them. Examine the diagram below, which demonstrates how a line segment AB is internally split by a point P(x, y) in a ratio such that AP: PB = m:n.

Now, for the provided coordinates of the points A(x1, y1), B(x2, y2), the coordinates of the point P in terms of the given ratio are represented as:

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

• The coordinates of point P are x and y.

• The coordinates of point A are (x1, y1).

• The coordinates of point B are (x2, y2)

• m and n are the ratio values by which P splits the line internally.

External Section Formula:

 The external section formula is used when the line segment is divided externally by the point in the provided ratio. This formula is used to get the coordinates of the point on the line segment connecting the two locations and falling beyond them in the provided ratio. Examine the diagram below, which illustrates that the line segment AB is externally divided by a point P(x, y) in a ratio such that AP: PB = m:n.

Now, for the provided coordinates of the points A(x1, y1), B(x2, y2), the coordinates of the point P in terms of the given ratio are represented as:

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

• The coordinates of point P are x and y.

• The coordinates of point A are (x1, y1).

• The coordinates of point B are (x2, y2)

•  m and n are the ratio values by which P splits the line externally.

Section Formula for Midpoint:

The midpoint formula is used to get the coordinates of the midpoint of a line segment. In this situation, the ratio between the two portions is 1:1. A midpoint is a point in the line segment that is exactly in the centre.

It can be written as taking m and n equal to 1 so that the required formula becomes:

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

Conclusion:

The Section formula is used in coordinate geometry to compute the ratio of a line segment split by a point, either internally or externally. It is used to determine the triangle’s centroid, incenter, and excentres. It is used in physics to locate the centre of mass of systems, equilibrium locations, and other things.