How to find the point ‘P’ that divides the line segment AB in a given ratio.

( Both internal and external division)

Goal:

Theory:

Part 1

(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:

Once again, the blue formula is for three-dimensional space, where you have ‘z’ coordinates as well as ‘x’ and ‘y’.

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Method:

Part 2

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 1 Download the free support file…

We have created a Maths Helper Plus document to complement this topic. You can use it to practice the steps described below, and as a starting point for solving your own problems.

File name: ‘Divide Line Segment.mhp’ File size: 7kb
Click here to download the file.

If you choose ‘Open this file from its current location’, then Maths Helper Plus should open the document immediately. If not, try the other option: ‘Save this file to disk’, then run Maths Helper Plus and choose the ‘Open’ command from the ‘File’ menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

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.

Divide AB internally

Calculations using ‘x’ and ‘y’ values…

 x          y

0.4     1.4

Divide AB externally

   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 labeled diagram

If the labeled 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.

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