## Section Formula

The Section formula is used to calculate the position information of the point which thus splits a linear function either externally or internally into a certain ratio. This formula is frequently used in math and physics.

Section formula has been used in math and science to locate the center point, in centers as well as ex centers of a triangle, and in science to find the center of gravity, optimum points, and so on. To discover the middle point of a vertical line, the section formula is commonly used.

## Use of Section Formula in Coordinate Geometry

The section technique is used to calculate the position information of a point on a vertical line that separates it into 2 parts. Assume we have such a point F(x,y) which thus divides a linear function marked as P(x

_{1},y_{1}) and Q(x_{2},y_{2}). Then we use the section process to determine the coordinates, which also is mathematically demonstrated as:

F (X, Y) = [(MX_{2 }+NX_{1})/(M+N) ,(MY_{2}+NY_{1})/(M+N)]

In coordinate geometry, the section formula is broken down into sub, and are as follows:

The formula for the internal section

The formula for the external section

## Solved Examples

**Example 1: Find the coordinates of the point that divides the line segment joining the points (2,5) and (-3, -10) internally in the ratio 3:2.**

**Solution: **Let F (x, y) be the point that divides the line segment joining A (2, 5) and B (-3, -10) internally in the ratio 3: 2.

Here,

(x

_{1}, y_{1}) = (2, 5)(x

_{2}, y_{2}) = (-3, -10)m: n = 3: 2

Using the section formula,

F (X, Y) = [(MX

_{2 }+ NX_{1})/(M+N) ,(MY_{2 }+ NY_{1})/(M+N)]

THEREFORE, THE COORDINATES OF F ARE

X= [(3× -3+2×2)/(3+2)] = -1

Y= [(3× -10+2×5)/(3+2)] = -4

THEREFORE F (X, Y) = (-1, -4)

**Example 2: Find the coordinates of the point that divides the line segment joining the points (2,5) and (-2, -1) internally in the ratio 2:2.**

**Solution:** Let F (x, y) be the point that divides the line segment joining A (2, 5) and B (-2, -1) internally in the ratio 2: 2.

Here,

(x

_{1}, y_{1}) = (2, 5)(x

_{2}, y_{2}) = (-2, -1)m : n = 2: 2

Using the section formula,

F (X, Y) = [(MX

_{2 }+ NX_{1})/(M+N) ,(MY_{2 }+ NY_{1})/(M+N)]

THEREFORE, THE COORDINATES OF F ARE

X= [(3× -2+2×2)/(2+2)]

Y= [(3× -1+2×5)/(2+2)]

THEREFORE F (X, Y) = (-0.5,1.5)