### 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 $$6.2 \ cm$$. Step 2: Draw any ray $$AX$$, making an acute angle with $$AB$$. 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 $$AA_1$$ $$= A_1A_2$$ $$= A_2A_3$$ $$= A_3A_4$$ $$= A_4A_5$$ $$= A_5A_6$$ $$= A_6A_7$$. Step 4: Join $$BA_7$$. 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 $$2.6 \ cm$$. 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 $$A_4C$$.

$\frac{A{A}_{4}}{{A}_{4}{A}_{7}}=\frac{\mathit{AC}}{\mathit{CB}}$

$\frac{A{A}_{4}}{{A}_{4}{A}_{7}}=\frac{4}{3}$ (by construction)

$\frac{\mathit{AC}}{\mathit{CB}}=\frac{4}{3}$

Therefore, $$AC:CB = 4: 3$$.