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The distances from the ends of a segment to a plane are 1 and 3. What can the distance from the midpoint of this segment to the same plane be?

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Let $A 1, B 1$ and $M 1$ be the orthogonal projections onto the plane $\alpha$ of points $A$ and $B$ and the midpoint $M$ of segment $A B$, respectively. Then the distance from point $M$ to the plane $\alpha$ is equal to the length of segment $M M 1$. If points $A$ and $B$ are on the same side of the plane $\alpha$, then $M M 1$ is the midline of the rectangular trapezoid $A A 1 B 1 B$ with bases $A A 1$ and $B B 1$. Therefore,

$$
M M 1=\frac{1}{2}(A A 1+B B 1)=\frac{1}{2}(1+3)=2
$$

If points $A$ and $B$ are on opposite sides of the plane $\alpha$, then $M M 1$ is the segment connecting the midpoints of the diagonals of the rectangular trapezoid $A A 1 B B 1$ with bases $A A 1$ and $B B 1$. Therefore,

$$
M M 1=\frac{1}{2}|A A 1-B B 1|=\frac{1}{2}(3-1)=1 .
$$

## Answer

2 or 1.