
Line segments
Given line segment $\displaystyle AB$,
1.) Where should one place point $\displaystyle C$ such that the ratio of the length of $\displaystyle AB$ to the length of $\displaystyle AC$ is $\displaystyle 2$? When $\displaystyle C$ is in this position, what's the ratio of the length of $\displaystyle AC$ to $\displaystyle BC$?
2.) Where should one place point $\displaystyle C$ such that the ratio of the length of $\displaystyle AB$ to the length of $\displaystyle AC$ is $\displaystyle 1$? Are you able to determine the ratio of $\displaystyle AC$ to $\displaystyle BC$? If so, find it. If not, explain why not.
3.) Using your answers from #1 and #2, determine the smallest and largest possible numbers for the ratio of the length of the whole to the length of the greater  that is, the length of the whole is AB and the greater will be the bigger line segment cut but C.

My solutions:
For #1:
I would put C in the middle this would make the ratio 2. The value of AC/BC would be 1/2.
For #2:
I would put C where B is. Then it's 1. Then AC/BC would be 1 too would it not? I think?
For #3: Not sure

OK for 1 but AC/BC=1
OK for 2 but B=C means BC=0 so ...