Task 1. The segments $A B, C D$ are marked on the right, for which $\frac{1}{4}$ of their lengths is common. Determine the lengths of the segments if the distance between their midpoints is equal to $6 \mathrm{~cm}$.
Solution. The segments share a quarter of their length, that is,

![](https://cdn.mathpix.com/cropped/2024_06_05_2e9ecab3ff55f9ab2430g-05.jpg?height=81&width=425&top_left_y=1673&top_left_x=1013)
they coincide on one quarter of their length, so we conclude that the two segments have equal lengths. In the diagram, the midpoints $M$ of $AB$ and $N$ of $CD$ are marked. We will denote one quarter of the length of the segments by $x$. Then, according to the diagram, for the distance between the midpoints we have $\overline{MN}=3x=6 \mathrm{~cm}$, so $x=2 \mathrm{~cm}$. Therefore, $\frac{1}{4}$ of the length of one segment is equal to $2 \mathrm{~cm}$, so $\overline{AB}=\overline{CD}=4x=8 \mathrm{~cm}$.