# Introduction to Three Dimensional Geometry, Coordinates of a Point of Division of a Line Segment

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## Coordinates of a Point of Division of a Line Segment

Let the point divide in the ratio internally.

Let the co-ordinates of be and the co-ordinates of be .

From points and , draw and perpendiculars to the -plane.

Draw and perpendiculars to .

Now and

Also we have,

Or

Or

Or

Similarly, if we draw perpendiculars to and respectively,

We get and

is the point

If , then the co-ordinates of the point which divides in the ratio are

It is clear that to every value of , there corresponds a point of the line and to every point on the line , there corresponds some value of .

If is postive, lies on the line segment and if is negative, does not lie on line segment.

In the second case you may say the divides the line segment externally in the ratio.

**Corollary 1**:

The co-ordinates of the point dividing externally in the ratio are

**Corollary 2**:

The co-ordinates of the mid-point of are

Example:

Find the co-ordinates of the point which divides the line segment joining the points and in the ratio internally.

Solution:

Let, be the two points.

Let divides in the ratio.

And

Thus, the co-ordinates of are .