If is the midpoint of the line joining
and
then
Hence
Convention:
In case of internal division:
is the midpoint of
.
Hence be a point on
, which divides
internally.
There are two conditions:
(i)
(ii)
Note:
(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:
and
Alternatively,
and
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
(Hence )
Some Example on the above Properties:
Distance Formula
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
Mid-Point Formula:
Let and
be end points of
and
be midpoint of
, then