Problem 4. Points $A, B, C, D, E$ are marked on a straight line in that order. The distance between the midpoints of segments $A B$ and $D E$ is $16 \mathrm{~cm}$, and the distance between the midpoints of segments $B C$ and $C D$ is $6 \mathrm{~cm}$. Calculate the length of segment $A E$.
Solution. We have, $\overline{B C}+\overline{C D}=\overline{B D}$. Furthermore, $\frac{1}{2} \overline{B C}+\frac{1}{2} \overline{C D}=6 \text{ cm}$, so

![](https://cdn.mathpix.com/cropped/2024_06_05_c3c23cf15e32fbc44a06g-03.jpg?height=72&width=522&top_left_y=1822&top_left_x=955)
$\overline{B D}=2 \cdot 6=12 \text{ cm}$. Similarly, $\frac{1}{2} \overline{A B}+\overline{B D}+\frac{1}{2} \overline{D E}=16 \text{ cm}$, from which we get $\frac{1}{2} \overline{A B}+\frac{1}{2} \overline{D E}=16-12=4 \text{ cm}$. Finally,

$$
\overline{A E}=16+\frac{1}{2} \overline{A B}+\frac{1}{2} \overline{D E}=16+4=20 \text{ cm}
$$

## VI Section