[ Measurement of lengths of segments and measures of angles. Adjacent angles.] $[\quad$ Extreme properties (other).

In a village along a straight road, three houses $A, B$, and $C$ are located at intervals of 50 meters.

At what point on the road should a well be dug so that the sum of the distances from the well to the houses is minimized?
Let point $B$ be located between $A$ and $C$. If point $B$ were not there, the well could be dug at any point on the segment $A C$ (see problem $\underline{54738}$).

Now let's add point $B$. It is clear now that the well should be dug at the house $B$.

## Answer

At the middle house.

In village $A$, 100 schoolchildren live, and in village $B$, 50 schoolchildren live. The distance between the villages is 3 kilometers.

At what point on the road from $A$ to $B$ should a school be built so that the total distance traveled by all schoolchildren is as small as possible?

## Solution

Let's temporarily forget about the 50 schoolchildren from $A$. Then the school can be built at any point on the segment $A B$: each pair of schoolchildren from $A$ and $B$ together travel 3 km. But for the remaining 50 schoolchildren from $A$, the total distance is minimized when the school is located in $A$.

## Answer

In village $A$.