What are the coordinates of the point on the directed line segment from (−1,−8) to (5,−2) that partitions the segment into a ratio of 1 to 2?
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Answer:

There are two possible options: [tex]P(x,y) = (1, -6)[/tex] or [tex]P(x,y) = (3, -4)[/tex].

Step-by-step explanation:

There are two possible options depending on what the point of origin is. Vectorially speaking, we can determine the coordinates of the point that partitions the segment is described below:

[tex]P(x,y) = A(x,y) + r\cdot [B(x,y) - A(x,y)][/tex] (1)

Where:

[tex]A(x,y)[/tex] -  Point of origin.

[tex]B(x,y)[/tex] - Point of destination.

[tex]r[/tex] - Partition factor.

Option 1: [tex]A(x,y) = (-1, -8)[/tex], [tex]B(x,y) = (5, -2)[/tex], [tex]r = \frac{1}{3}[/tex]

[tex]P(x,y) = (-1, - 8) + \frac{1}{3}\cdot [(5,-2)-(-1,-8)][/tex]

[tex]P(x,y) = (-1, -8) + \frac{1}{3}\cdot (6,6)[/tex]

[tex]P(x,y) = (-1,-8) + (2,2)[/tex]

[tex]P(x,y) = (1, -6)[/tex]

Option 2: [tex]A(x,y) = (5,-2)[/tex], [tex]B(x,y) = (-1,-8)[/tex], [tex]r = \frac{1}{3}[/tex]

[tex]P(x,y) = (5,-2) + \frac{1}{3}\cdot [(-1,-8)-(5,-2)][/tex]

[tex]P(x,y) = (5,-2) +\frac{1}{3}\cdot (-6,-6)[/tex]

[tex]P(x,y) = (5,-2) +(-2,-2)[/tex]

[tex]P(x,y) = (3, -4)[/tex]

There are two possible options: [tex]P(x,y) = (1, -6)[/tex] or [tex]P(x,y) = (3, -4)[/tex].

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