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.
 
fig_1.svg
 
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, APPB=mn.
 
xx1x2x=mn
 
\(n(x - x_1)\) \(=\) \(m(x_2 - x)\)
 
\(nx - nx_1 = mx_2 - mx\)
 
\(mx + nx = mx_2 + nx_1\)
 
\(x(m + n) = mx_2 + nx_1\)
 
x=mx2+nx1m+n
 
If \(A\), \(P\), and \(B\) has the coordinates \((x_1\), \(y_1)\), \((x\), \(y)\), and \((x_2\), \(y_2)\) respectively, then:
 
x=mx2+nx1m+n
 
y=my2+ny1m+n