# 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.

1. 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:

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

1. $$P(6.8,5.4)$$
2. $$P(3.8,8.4)$$

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