Theory:

Let's imagine you're given a line segment, and you need to divide it in a specific ratio, say \(2:3\). You can achieve this by measuring the length and then marking a point on it that splits it in the ratio specified. But what if you don't have a way to measure it accurately? How would you determine the point? We will see how to locate these points through example.
To divide a line segment in a given ratio
Example:
Draw a line segment of length \(6.2 \ cm\) and divide it in the ratio \(4:3\). Measure the two parts.
 
Step 1: Draw a line segment \(AB\) of length \(6.2 \ cm\).
 
Pic1.png
 
Step 2: Draw any ray \(AX\), making an acute angle with \(AB\).
 
Pic2.png
 
Step 3: Mark \(4 + 3 = 7\) points \(A_1\), \(A_2\), \(A_3\), \(A_4\), \(A_5\), \(A_6\) and \(A_7\) on \(AX\) such that \(AA_1\) \(= A_1A_2\) \(= A_2A_3\) \(= A_3A_4\) \(= A_4A_5\) \(= A_5A_6\) \(= A_6A_7\).
Pic3.png
 
Step 4: Join \(BA_7\).
 
Pic4.png
 
Step 5: Draw a line parallel to \(BA_7\) at \(A_4\) intersecting the line segment \(AB\) at \(C\). Thus, \(AC: CB = 4:3\). The length of the parts are \(3.6 \ cm\) and \(2.6 \ cm\).
 
Pic5.png
 
JUSTIFICATION:
 
In the figure, \(AX\) is a transversal for lines \(A_7B\) and \(A_4B\).
 
 \(\angle AA_7B = \angle AA_4C\)
 
If corresponding angles are equal, then the lines are parallel.
 
Therefore, \(A_7B\) is parallel to \(A_4C\).
 
 
AA4A4A7=ACCB
 
AA4A4A7=43 (by construction)
 
ACCB=43
 
Therefore, \(AC:CB = 4: 3\).