Problem 2.1. Points $A, B, C, D$ are marked on a line, in that exact order. Point $M$ is the midpoint of segment $A C$, and point $N$ is the midpoint of segment $B D$. Find the length of segment $M N$, given that $A D=68$ and $B C=20$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2a01f5e31e9fc3823579g-06.jpg?height=210&width=832&top_left_y=323&top_left_x=315)

## 68
Answer: 24.

Solution. Let $A C=x$, then $A M=\frac{x}{2}$. Now we calculate the length of $N D$:

$$
N D=\frac{B D}{2}=\frac{20+C D}{2}=\frac{20+(68-x)}{2}=44-\frac{x}{2}
$$

Now it is not difficult to calculate $M N$:

$$
M N=A D-A M-N D=68-\frac{x}{2}-\left(44-\frac{x}{2}\right)=24
$$