Introduction to section formula — lesson. Mathematics State Board, Class 9.

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Until now, we have only dealt with mid-point and the points of trisection.

Mid-point divides the line segment into two halves and the points of trisection divides the line segment into

$3$ equal parts.

But, is it possible to divide the line segment into two unequal parts?

Yes, a line segment can be divided into two unequal parts using the section formula.

Imagine you have

$6$ milk packets and two bags of unequal sizes.

Bag

$A$ can hold

$4$ milk packets while bag

$B$ can hold only

$2$ milk packets.

In this case, a total of

$6$ milk packets is distributed across the two bags in the ratio of

$4:2$ .

Similarly, a line segment can also be divided in unequal ratios.

Let us look at how a section formula gets constructed.

In the figure given above, a line segment

$AB$ is divided into two unequal parts in the ratio

$m : n$ .

Let

$A$ be

$x_1$ ,

$P$ be

$x$ and

$B$ be

$x_2$ such that

$x_2 > x > x_1$ .

The co-ordinate of

$P$ divides the line segment in the ratio

$m : n$ .

This means,

$\frac{AP}{PB} = \frac{m}{n}$

$\frac{x - x_1}{x_2 - x} = \frac{m}{n}$

$m(x_2 - x)$

$=$

$n(x - x_1)$

$mx_2 - mx = nx - nx_1$

$mx_2 + nx_1 = mx + nx$

$x$

$=$

$\frac{mx_2 + nx_1}{m + n}$

$A$ ,

$P$ , and

$B$ has the co-ordinates

$(x_1$ ,

$y_1)$ ,

$(x$ ,

$y)$ , and

$(x_2$ ,

$y_2)$ respectively, then:

$x$

$=$

$\frac{mx_2 + nx_1}{m + n}$

$y$

$=$

$\frac{my_2 + ny_1}{m + n}$

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1. Introduction to section formula

Theory: