Find the coordinates of point $P$ along the directed line segment $AB$ so that $AP$ to $PB$ is the given ratio. $A\left(-4,\ -8\right)$ , $B\left(0,\ 0\right)$ ; 3 to 1

Answer 1

The formula is

[(b*x1 + a*x2)/(a+b) , (b*y1+a*y2)/(a+b)]

A=(-4,-8) is left end point

B=(0,0) is right end point

splits line segment in proportion 3 to 1

x1=-4 x2 = 0 y1 = -8 and y2=0

a=3 and b=1 ---> a+b=4

(1*-4 + 3*0)/4 --> -1

(1* -8 + 3*0)/4 ---> -2

X(-1,-2) splits the line segment

from A(-4,-8) to B(0,0) in the ratio

3 to 1

AX = sqrt ( (-4 - -1)^2 + (-8 - -2)^2)

= sqrt ( (-3)^2 + (-6)^2)

= sqrt ( 9 + 36)

= sqrt(45)

= 3 *sqrt(5)

BX = sqrt ( 1 + 4) = sqrt(5)

AB = sqrt(16+64) = sqrt(80) = 4*sqrt(5)

AX/AB = 3/4 and BX/AB = 1/4, so yes it checks

The official answer is (-1,-2)