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# Division of an interval in a given ratio

The division of an interval in a given ratio involves finding the coordinates of a point that divides the line joining two points A(x1, y1) and B(x2, y2) in the given ratio of m:n.

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)

x  =

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

So   =

m(y2 – y)  =  n(y – y1)

my2 – my  =  ny – ny1

my2 + ny1 – my  =  ny

my2 + ny1  =  ny + my

my2 + ny1  =  y(n + m)

y  =

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

y  =

Let’s look at an example of division of an interval in a given ratio.

Note:

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.