Question

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?

I need help quick

Answer 1

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

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:

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

Where:

`A(x,y)` - Point of origin.

`B(x,y)` - Point of destination.

`r` - Partition factor.

Option 1:
`A(x,y) = (-1, -8)`,
`B(x,y) = (5, -2)`,
`r = (1)/(3)`

`P(x,y) = (-1, - 8) + (1)/(3)\cdot [(5,-2)-(-1,-8)]`

`P(x,y) = (-1, -8) + (1)/(3)\cdot (6,6)`

`P(x,y) = (-1,-8) + (2,2)`

`P(x,y) = (1, -6)`

Option 2:
`A(x,y) = (5,-2)`,
`B(x,y) = (-1,-8)`,
`r = (1)/(3)`

`P(x,y) = (5,-2) + (1)/(3)\cdot [(-1,-8)-(5,-2)]`

`P(x,y) = (5,-2) +(1)/(3)\cdot (-6,-6)`

`P(x,y) = (5,-2) +(-2,-2)`

`P(x,y) = (3, -4)`

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