Question

What are the coordinates of the point on the directed line segment from (-5, – 7) to(1, -1) that partitions the segment into a ratio of 1 to 2?

Answer 1

The rule of the coordinates of the point which divides a line whose endpoints are (x1, y1) and (x2, y2) at a ratio m: n from the first point is

`\begin{gathered} x=(x_2m+x_1n)/(m+n) \\ y=(y_2m+y_1n)/(m+n) \end{gathered}`

Since the endpoints of the directed line are (-5, -7) and (1, -1), then

`\begin{gathered} x_1=-5,x_2=1 \\ y_1=-7,y_2=-1 \end{gathered}`

Since the point (x, y) divide the line at a ratio of 1: 2, then

`m=1,n=2`

Substitute them in the rule above

`\begin{gathered} x=((1)(1)+(-5)(2))/(1+2) \\ x=(1-10)/(3) \\ x=(-9)/(3) \\ x=-3 \end{gathered}`
`\begin{gathered} y=((-1)(1)+(-7)(2))/(1+2) \\ y=(-1-14)/(3) \\ y=(-15)/(3) \\ y=-5 \end{gathered}`

The coordinates of the point are (-3, -5)

The answer is (-3, -5)

The length of the line from point (-5, -7) to point (1, -1) is 6 diagonals of the small square

We will divide it by the sum of the ratio ( 1 + 2 = 3)

Divide 6 by 3, then each part = 2

Then the distance from the point (-5, -7) to the point of division is 2 diagonals of the small squares

Then the point of the division lies on (-3, -5)