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
[tex]\begin{gathered} x=\frac{x_2m+x_1n}{m+n} \\ y=\frac{y_2m+y_1n}{m+n} \end{gathered}[/tex]Since the endpoints of the directed line are (-5, -7) and (1, -1), then
[tex]\begin{gathered} x_1=-5,x_2=1 \\ y_1=-7,y_2=-1 \end{gathered}[/tex]Since the point (x, y) divide the line at a ratio of 1: 2, then
[tex]m=1,n=2[/tex]Substitute them in the rule above
[tex]\begin{gathered} x=\frac{(1)(1)+(-5)(2)}{1+2} \\ x=\frac{1-10}{3} \\ x=\frac{-9}{3} \\ x=-3 \end{gathered}[/tex][tex]\begin{gathered} y=\frac{(-1)(1)+(-7)(2)}{1+2} \\ y=\frac{-1-14}{3} \\ y=\frac{-15}{3} \\ y=-5 \end{gathered}[/tex]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)
