[ Measuring the lengths of segments and the measures of angles. Adjacent angles. ]

Points $M, A$ and $B$ are located on the same line, and segment $A M$ is twice as long as segment $B M$. Find $A M$ if $A B=6$.
Consider all possible cases of the arrangement of points $A, B$, and $M$.

## Solution

If point $M$ lies between points $A$ and $B$, then $A M = 2 / 3 A B = 4$.

If point $B$ lies between points $A$ and $M$, then $B$ is the midpoint of $A M$, so $A M = 2 A B = 12$.

Point $A$ cannot lie between points $B$ and $M$, as in this case the segment $A M$ would be shorter than the segment $B M$.

## Answer

4 or 12.