The Art of Partitioning a Line Segment!

To partition a line segment in a given ratio, find the total number of parts, work out what fraction of the way along the point falls, and apply that fraction to the change in x and the change in y from the starting endpoint.

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} )

Worked examples: partitioning a line segment

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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