The Art of Partitioning a Line Segment!

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

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. \(P(3.8,8.4)\)

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