determine the point p that partitions the directed line segment mn into a ratio of 2:3, where m (3,5) and n(8,-5)

Question

determine the point p that partitions the directed line segment mn into a ratio of 2:3, where m (3,5) and n(8,-5)

Answer 1

This is an application of the so-called Side-splitter Theorem from plane geometry.

If the co-ordinates of a ends of a line segment are A(a,b) and B(c,d) and you want to find a point 100x% of the way from A to B, the co-ordinates of that point are

[(1-x)a+xc,(1-x)b+xd).

In this problem x=.4

Answer 2

The x coordinate will be 3 + (2/2+3) x (8-3)

The y coordinate will be 5 + (2 / 2+3) x (-5-5)

In general if the ratio is a:b, and the points (x1,y1) (x2,y2) will be

x : x1 + (a/[a+b]) (x2-x1)

y y1 + (a/[a+b]) (y2-y1)