If is the midpoint of the line joining and then
In case of internal division:
is the midpoint of .
Hence be a point on , which divides internally.
There are two conditions:
(i) We write divides or the line segment joining and in ratio to mean that .
(ii) For divides or the line segment joining and in ratio to mean that .
In Case of External Division:
Coordinates of any Points on the line:
If and be the coordinates of divides in the ratio (Either internally or externally). Then it can be expressed as or where,
For Internal Division:
For External Division
Some Example on the above Properties:
Coordinates of and are and respectively, then
Distance from Origin
Let be a point on a Cartesian Plane then
Internal Division on Ratio
Let and be the end points of a line segment . divides internally in ratio , the coordinate of is
Internal Division in Ratio
Let and be the end points of . divides in ratio internally the coordinate of is
External Division in ratio
Let and be two end points on . Let be a point divides externally in ratio , then
External Division in Ratio
Let and be the two end points of . Let be a point divides in ratio , externally, then
Let and be end points of and be midpoint of , then