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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.division of an interval

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 DeltaABG, PF parallel BG, and divides side AB and AG in the same ratio (intersect property of parallel lines)

AP/PB  =  AF/FG

But  AP/PB  =  m/n (given)

Hence  m/n  =  AF/FG  =  {~x~-~x_1~}/{~x_2~-~x~}

Since AF = x – x1, and FG = x2 – x

m/n  =  {~x~-~x_1~}/{~x_2~-~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  =  {~mx_2~+~nx_1~}/{m~+~n}

Similarly we can draw lines AJ, PI and BH perpendicular to y-axis and we get

AP/PB  =  IJ/HI  (since BH parallel PI, using the intercept property)

m/n  =  IJ/HI

HI = y2 – y and IJ = y – y1

So m/n  =  {~y~-~y_1~}/{~y_2~-~y~}

m(y2 – y)  =  n(y – y1)

my2 – my  =  ny – ny1

my2 + ny1 – my  =  ny

my2 + ny1  =  ny + my

my2 + ny1  =  y(n + m)

y  =  {~my_2~+~ny_1~}/{m~+~n}

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  =  {~mx_2~+~nx_1~}/{m~+~n}  and

y  =  {~my_2~+~ny_1~}/{m~+~n}

 

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.

internal division

 

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.

external division_1

external division_2

 

Here are some examples of division of an interval.