Distance/midpoints

Question

If point k is 1/n of the way from J (4,-5) to L(0.-7)

n=4

what is a formula for the coordinates of k for any m?

Answer 1

You can think of the coordinates of K as a weighted average of the J and L coordinates. If n = 1 then K is 100% of the way from J to L, in other words exactly at L. As n gets larger, say n = 10, then K is 10% of the way from J to L, in other words 90% of J plus 10% of L.

For your example n = 4. So the coordinates of K are:

x = (1-1/4)*4 + (1/4)*0 = 3

y = (1-1/4)*-5 + (1/4)*-7 = -5.5

K: (3, -5.5)

In general:

x = (1-1/n)*4 + (1/n)*0

y = (1-1/n)*-5 + (1/n)*-7

Answer 2

JL in xy plane can be seen as hypotenuse

of right triangle. If point is 1/n from J to L

then point's x coordinate is 1/n from x_J to x_l

(similar triangle theorem)

x_1/n=((x_L-x_J)/n)+x_k=((x_L-x_J)/n)+nx_k/n

x_1/n=[(n-1)x_J+x_L]/n

for n=4

x_1/4=[3x_J+x_L]/4

similarly

y_1/n=[(n-1)y_J+y_L]/n

y_1/4=[3y_J+y_L]/4

for the p/q point on JL ,,,,,,p/q<1

using same formulazation

x_p/q=[(q-p)x_J+px_L]/q

y_p/q=[(q-p)y_J+py_L]/q