### Theory:

We have learnt how to plot the points on a coordinate plane and find the distance between two points.

Now, we will learn how to divide a line segment into two parts.

Do you think 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 into 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, $\frac{\mathit{AP}}{\mathit{PB}}=\frac{m}{n}$.

$\frac{x-{x}_{1}}{{x}_{2}-x}=\frac{m}{n}$

$$n(x - x_1)$$ $$=$$ $$m(x_2 - x)$$

$$nx - nx_1 = mx_2 - mx$$

$$mx + nx = mx_2 + nx_1$$

$$x(m + n) = mx_2 + nx_1$$

$x=\frac{m{x}_{2}+n{x}_{1}}{m+n}$

If $$A$$, $$P$$, and $$B$$ has the coordinates $$(x_1$$, $$y_1)$$, $$(x$$, $$y)$$, and $$(x_2$$, $$y_2)$$ respectively, then:

$x=\frac{m{x}_{2}+n{x}_{1}}{m+n}$

$y=\frac{m{y}_{2}+n{y}_{1}}{m+n}$