Problem 4. A segment is divided by three points into four unequal parts. The distance between the midpoints of the inner parts is $6 \mathrm{~cm}$, and the distance between the midpoints of the outer parts is $16 \mathrm{~cm}$. What is the length of the given segment?
Solution. Let the endpoints of the given segment be $A$ and $B$, and let the points $C, D, E$ divide it into four unequal parts. Let $M$ be the midpoint of

![](https://cdn.mathpix.com/cropped/2024_06_05_dc4d34c0a9dd2a3ba572g-04.jpg?height=113&width=639&top_left_y=1155&top_left_x=570)

$AC$, $N$ the midpoint of $CD$, $P$ the midpoint of $DE$, and $Q$ the midpoint of $EB$. Then, from the condition $\overline{NP}=6 \text{ cm}$ and $\overline{MQ}=16 \text{ cm}$ (see the diagram). Now, from $\overline{NP}=6 \text{ cm}$, we have that $\overline{CD}+\overline{DE}=2 \cdot 6=12 \text{ cm}$. From this and from $\overline{MQ}=16 \text{ cm}$, we get that $\overline{MC}+\overline{EQ}=16-12=4 \text{ cm}$, and since $M$ is the midpoint of $AC$ and $Q$ is the midpoint of $EB$, it follows that $\overline{AM}+\overline{QB}=4 \text{ cm}$. Finally, the length of the given segment is $\overline{AB}=(\overline{AM}+\overline{QB})+\overline{MQ}=4+16=20 \text{ cm}$.