Find all possible locations for point (geometry)

Question

JK in the coordinate plane has endpoints with coordinates (-4,11) and (8, -1). Find all possible locations for point M so M divides JK into two parts with lengths in a ratio of 1:3.

possible answers :

(-1,8)

(2,5)

(5,2)

(8,-1)

(-4,11)

Answer 1

If the co-ordinates of the endpoints of a line segment are A(a,b) and B(c,d) and you want to find the co-ordinates of a point 100p% of the way from A to B, the co-ordinates are

[(1-p)a+pc, (1-p)b+pd].

In your problem p=.25.

BTW, this is an application of the so-called Side-splitter Theorem of palne geometry.