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This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to find the point 'P' that divides the line segment AB in a given ratio. It covers both internal and external division. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

(a) Internal division of a line segment:

Consider the point P(x_P , y_P) that lies on the line segment AB joining points A(x_A , y_B) and B(x_A , y_B):

Point P divides AB in the ratio k₁ : k₂ , and the coordinates of point P are given by these formulas:

The blue formula is for three dimensional space, where you have 'z' coordinates as well as 'x' and 'y'.

When point P lies between points A and B, it divides AB internally.

(b) External division of a line segment:

When point P lies outside of the line segment AB, it can divide AB externally as shown in this diagram:

The formulas for external division are as follows:

Method:

Maths Helper Plus can substitute values into the line division formulas and so calculate the point 'P' for both internal and external division. Also, it will create a diagram showing the original points A and B, as well as the internal and external division points.  

 

Step 2  Enter points A and B

To enter your own points 'A' and 'B', replace the existing points: A(-2,3) and B(4,-1).

First click on the text view on the coordinate you want to change. (See diagram below.) 

                 

Make your changes, then press the 'Enter' key.

 

Step 3  Enter the ratio numbers to divide AB

Press the F5 key to display the parameters box:

 

Enter the 'k₁' and 'k₂' values are entered into the 'C' and 'D' edit boxes. 

For example, if k₁ = 2 and k₂ = 3, then enter C = 2 and D = 3.

 

To enter the values, click on the 'C' edit box, type your k₁ value, then click on the 'D' edit box and type the k₂ value. Now click the Update button.

 

The coordinates of the dividing point 'P' are displayed on the text view as follows:

 

        x     y

        0.4   1.4

        x     y

        -14   11

 

Also, the graph view displays a diagram showing the plotted points A and B, as well as the dividing points for internal and external division. The internal division point is purple, while the external division point is green.

 

Step 4  Adjust the scale of the labelled diagram

If the labelled diagram does not display all of the points, the scale needs to be reduced. For the external division point this is most likely as it is often far from both A and B. In this case, briefly press the F10 key enough times until the required points are seen.

 

You can also enlarge the diagram again by holding down 'Ctrl' while you press F10.