# The Art of Partitioning a Line Segment!

In geometry, the simple act of drawing a line segment can lead to deeper, fascinating explorations. One such exploration is the partitioning of a line segment. Imagine dividing a chocolate bar into pieces, ensuring each friend gets an equal share, or maybe one gets twice as much as another. Similarly, in geometry, we can split a line segment into multiple parts based on a specific ratio. Let's delve into the process of partitioning a line segment and understand its mathematical underpinnings.

## Step-by-step Guide: Partitioning a Line Segment

**Understanding the Concept**:

Partitioning a line segment involves dividing it into multiple parts, where each part is a fraction or multiple of the whole segment. This can be done based on a given ratio.

**Mathematical Representation**:

Given a line segment \(AB\), we can partition it at a point \(P\) such that the ratio of \(AP\) to \(PB\) is \(m:n\), where \(m\) and \(n\) are positive integers.

**Formula for Partitioning**:

If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the endpoints of the segment, and we want to partition the segment in the ratio \(m:n\), the coordinates \((x, y)\) of point \(P\) are given by:

\( x = \frac{mx_2 + nx_1}{m+n} \)

\( y = \frac{my_2 + ny_1}{m+n} \)

### Examples

**Example 1**

Given the line segment with endpoints \(A(1,2)\) and \(B(7,8)\), find the point that partitions the segment in the ratio \(2:3\).

**Solution**:

Using the formula:

\( x = \frac{2 \times 7 + 3 \times 1}{2+3} = \frac{17}{5} = 3.4 \)

\( y = \frac{2 \times 8 + 3 \times 2}{2+3} = \frac{22}{5} = 4.4 \)

Thus, the required point is \(P(3.4, 4.4)\).

**Example 2**:

Partition the line segment with endpoints \(C(3,4)\) and \(D(9,12)\) in the ratio \(1:4\).

**Solution**:

Applying the formula:

\( x = \frac{1 \times 9 + 4 \times 3}{1+4} = \frac{21}{5} = 4.2 \)

\( y = \frac{1 \times 12 + 4 \times 4}{1+4} = \frac{28}{5} = 5.6 \)

The partition point is \(P(4.2, 5.6)\).

**Practice Questions**:

- For the line segment with endpoints \(E(2,3)\) and \(F(10,7)\), determine the point that partitions the segment in the ratio \(3:2\).
- Partition the line segment with endpoints \(G(-1,2)\) and \(H(5,10)\) in the ratio \(4:1\).

**Answers**:

- \(P(6.8,5.4)\)
- \(P(3.8,8.4)\)

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