The Art of Partitioning a Line Segment!

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 \(4:1\).

Answers:

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