[ Measurement of segment lengths and angle measures. Adjacent angles.]

On a straight line, points $A, B$, and $C$ are given. It is known that $A B=5$, and segment $A C$ is one and a half times longer than $B C$. Find the segments $A C$ and $B C$.
Consider all possible cases of the arrangement of points $A, B$, and $C$.

## Solution

Let point $B$ lie on the segment $A C$. Then $3 / 2 B C = A C = A B + B C = 5 + B C$, from which $B C = 10, A C = 15$.

Point $A$ cannot lie between points $B$ and $C$, as in this case $A C < B C$.

Let point $C$ lie between $A$ and $B$. Then $\frac{5}{2} B C = A C + B C = A B = 5$, from which $B C = 2, A C = 3$.

## Answer

$B C = 10, A C = 15$ or $B C = 2, A C = 3$.