# Cartesian System of Rectangular Co-Ordinates, Section Formula, Internal Division, External Division

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## Internal Division

Let and be two given points on a line and divide internally in the ratio.

To find: The coordinates and of point.

Construction: Draw and perpendiculars to from and respectively and and lie on. Also draw and .

Method: divides internally in the ratio. The Coordinates X and Y of Point RThe coordinates x and y of point R

lies on and

Also, in triangles, and ,

(Corresponding angles as )

And

(AAA similarity)

Also,

From (1), we have

and

and

Thus, the coordinates of are:

Coordinates of the mid-point of a line segment

If is the mid point of , then,

(As divides in the ratio )

Coordinates of the midpoint are

## External Division

Let divide externally in the ratio

To find: The coordinates of .

Construction: Draw and perpendiculars to from and respectively and and .

Method:

Clearly,

Or

and

These give:

and

Hence, the coordinates of the point of external division are

Example:

Find the coordinates of the point which divides the line segment joining the points) and internally and externally in the ratio.

Solution:

Let be the point of internal division.

and

has coordinates

If is the point of external division, then

and

Thus, the coordinates of the point of external division are

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