Question

What are the lengths of the segments into which the y-axis divides the segment joining (-6,-6) and (3,6) ?

Answer 1

make a sketch
label the intersection withe the y-axis as (0,y)
I have two similar triangles and by ratios
(6-y)/3 = (6+y)/6
18+3y = 36-6y
9y=18
y=2

so the sides of the smaller right-angled triangle are 3 and 4, and I recognize the 3, 4, 5 triangle
the other triangle is 6 , 8, 10

So the length of the two segments you want are 10 and 5.

Long way:
Find the equation of the line joining the two points.
Find the y-intercept of the line
Find the length of the two segments using the distance formula between two points