
How to
find the point 'P' that divides the line segment AB in a given ratio.
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 k_{1} : k_{2} , 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.
Â Step 2Â Enter points A and BTo 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.)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Make your changes, then press the 'Enter' key. Â Step 3Â Enter the ratio numbers to divide ABPress the F5 key to display the parameters box: Â Enter the 'k_{1}' and 'k_{2}' values are entered into the 'C' and 'D' edit boxes.Â For example, if k_{1} = 2 and k_{2} = 3, then enter C = 2 and D = 3. Â To enter the values, click on the 'C' edit box, type your k_{1} value, then click on the 'D' edit box and type the k_{2} value. Now click the Update button. Â The coordinates of the dividing point 'P' are displayed on the text view as follows: Â Â Â Â Â Â Â Â Divide AB internally Â Â Â Â Â Â Â Calculations using 'x' and 'y' values... Â Â Â Â Â Â Â xÂ Â Â Â y Â Â Â Â Â Â Â 0.4Â Â 1.4 Â Â Â Â Â Â Â Divide AB externally Â Â Â Â Â Â Â Calculations using 'x' and 'y' values... Â Â Â Â Â Â Â xÂ Â Â Â y Â Â Â Â Â Â Â 14Â Â 11 Â 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. Â Step 4Â Adjust the scale of the labelled diagramIf 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. Â You can also enlarge the diagram again by holding down 'Ctrl' while you press F10. Â
