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

Answer:

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

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:

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

Where:

$A(x,y)$ -  Point of origin.

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

$r$ - Partition factor.

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

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

$P(x,y) = (-1, -8) + \frac{1}{3}\cdot (6,6)$

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

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

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

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

$P(x,y) = (5,-2) +\frac{1}{3}\cdot (-6,-6)$

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

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

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