In the diagram here, we have to find the coordinates of the point P(x, y) that divides the line joining A(x1, y1) and B(x2, y2) in the given ratio of m:n.
Let’s start by drawing lines AD, PE and BC perpendicular to x-axis, and the line AFG is perpendicular to BC.
In ABG, PF BG, and divides side AB and AG in the same ratio (intersect property of parallel lines)
But = (given)
Hence = =
Since AF = x – x1, and FG = x2 – x
m(x2 – x) = n(x – x1)
mx2 – mx = nx – nx1
mx2 + nx1 – mx = nx
mx2 + nx1 = nx + mx
mx2 + nx1 = x(n + m)
Similarly we can draw lines AJ, PI and BH perpendicular to y-axis and we get
= (since BH PI, using the intercept property)
HI = y2 – y and IJ = y – y1
m(y2 – y) = n(y – y1)
my2 – my = ny – ny1
my2 + ny1 – my = ny
my2 + ny1 = ny + my
my2 + ny1 = y(n + m)
Hence the coordinate of a point P(x, y) which divides the interval joining A(x1, y1) and B(x2, y2) in the ratio of m:n is given by
x = and
Let’s look at an example of division of an interval in a given ratio.
If AP and PB are in the same direction, then m:n is positive, i.e. P lies within AB. In this case, we say P divides AB internally.
If AP and PB are measured in the opposite directions, then m:n is negative and P lies outside AB, either closer to A or B (as shown in the diagram). In that case, we say P divides AB externally. Here the ratios would be m:-n or -m:n for the two cases in the diagram.
Here are some examples of division of an interval.