Find the coordinates of point R that lies along the directed line segment from J(10,-5) to K(-2,-3)

Answer 1

J(10, -5) to K(-2,-3)

are points with a line through them with slope (-3+5)/(-2-10) = 2/-12 = -1/6

It's slightly downward sloping, almost horizontal

the equation of the line through J and K is

-1/6 = (y+3)/(x+2)

y+3 = (-1/6)x -2/6

y = (-1/6)x -1/3-3

y = (-1/6)x -10/3

R is a point (x,y) that is a solution to that equation

where -2<x<10 and -3<y<-5

If R is the midpoint of JK

then R is ((10-2)/2, (-5-3)/2))

R= (4, -4)

-4 = (-1/6)(4) - 10/3= -2/3-10/3 =-12/3 =-4

R = (4,-4) is the only point on JK that has integer values for both x and y coordinates

other points on the line segment with integer x coordinates are

(-1, -3 1/6)

(0, -3 1/3)

(1, -3 1/2)

(2, -3 2/3)

(3, -3 5/6)

(4, -4)

(5, -4 1/6)

(6, - 4 1/3)

(7, -4 1/2)

(8, - 4 2/3)

(9, - 4 5/6)

y decreases by 1/6 as x increases by 1

Plot the two ponts (-2, -3) and (10,-5)

connect them with a straight line

all the above points are on that line segment JK

for R(4,-4) JR = RK with R 1/2 way from J to K

for R(7, -4 1/2) JR = 3RK with R 1/4 of the way from J to K

for R(1, -3 1/2) RK=3JR with R 3/4 of the way from J to K

for R(0, -3 2/3) RK= 6JR

for R(-1, - 3 5/6) RK = 12JR

for R(9, -4 5/6) JR = 12JK

for R(8, -4 2/3) JR = 6JK

if you know the ratio of JR to JK you can determine the point R

if you know one coordinate of R, you can determine the other coordinate

without more information, there are an infinite number of possible coordinates for R