4. On the line segment $\overline{A B}$ of length $80 \mathrm{~cm}$, points $C, D$ and $E$ are marked in sequence such that point $C$ is closest to point $A$. The distance between the midpoints of segments $\overline{C D}$ and $\overline{D E}$ is $23 \mathrm{~cm}$. How many centimeters are the midpoints of segments $\overline{A C}$ and $\overline{E B}$ apart?
4. On the line segment $\overline{A B}$ of length $80 \, \text{cm}$, points $C, D$, and $E$ are marked in sequence such that point $C$ is closest to point $A$. The distance between the midpoints of segments $\overline{C D}$ and $\overline{D E}$ is $23 \, \text{cm}$. How many centimeters are the midpoints of segments $\overline{A C}$ and $\overline{E B}$ apart?

## Solution.

On the line segment $\overline{A B}$, let's mark the midpoints $F$ and $G$ of segments $\overline{C D}$ and $\overline{D E}$, and denote the equal lengths of segments $\overline{C F}$ and $\overline{F D}$, as well as the equal lengths of segments $\overline{D G}$ and $\overline{G E}$:

$a = |C F| = |F D|$ and $b = |D G| = |G E|$.

$1 \text{ POINT}$

Let's also denote the lengths of the segments given in the problem.

On the line segment $\overline{A B}$, let's mark the midpoints of segments $\overline{A C}$ and $\overline{E B}$ (for example, $P_{1}$ and $P_{2}$), and denote the equal lengths of segments $\overline{A P_{1}}$ and $\overline{P_{1} C}$, as well as the equal lengths of segments $\overline{E P_{2}}$ and $\overline{P_{2} B}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_5cfeb7f7e539b150a939g-09.jpg?height=411&width=1014&top_left_y=891&top_left_x=247)

(diagram) 1 POINT

We need to determine the distance between points $P_{1}$ and $P_{2}$, i.e., $c + 2a + 2b + d$. $1 \text{ POINT}$

Since $|F G| = a + b = 23 \, \text{cm}$,

it follows that $|C E| = 2a + 2b = 2(a + b) = 46 \, \text{cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_5cfeb7f7e539b150a939g-09.jpg?height=465&width=962&top_left_y=1498&top_left_x=250)

(calculation or annotated diagram) 1 POINT

The difference $80 \, \text{cm} - 46 \, \text{cm} = 34 \, \text{cm}$ corresponds to the sum of the lengths of segments $\overline{A C}$ and $\overline{E B}$, $1 \text{ POINT}$ i.e., $2c + 2d = 34 \, \text{cm}$, $1 \text{ POINT}$ from which it follows that $c + d = 17 \, \text{cm}$. 1 POINT

Finally, we can calculate the desired length:

$\left|P_{1} P_{2}\right| = c + 2a + 2b + d$

$\left|P_{1} P_{2}\right| = (c + d) + (2a + 2b) = 17 \, \text{cm} + 46 \, \text{cm} = 63 \, \text{cm}$.

1 POINT

TOTAL 10 POINTS

Note: The distance between the midpoints of segments $\overline{A C}$ and $\overline{E B}$ can also be calculated as:

$\left|P_{1} P_{2}\right| = |A B| - (c + d) = 80 \, \text{cm} - 17 \, \text{cm} = 63 \, \text{cm}$.