Of all the line segments connecting a point outside the straight line and a point on the straight line, the vertical line segment is the shortest, and so on, we call it the shortest path problem.

Among the connecting lines between two points, the line segment is the shortest.

As the picture shows, there are two villages, A and B, on both sides of the A River. Now a bridge will be built over the river. In order to facilitate traffic and make the sum of the distance between the bridge and the two villages the shortest, which point should be built on the river to meet the requirements? (Draw a picture and explain)

As shown in the figure, connecting AB intersection line A to point P, the sum of the distances from the bridge to these two villages is the shortest.

The line segment between two points is the shortest.

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Solving the problem of maximum distance difference by axisymmetric method

As shown in the figure, points A and B are on both sides of the straight line L. Find a point C on the line L to maximize the distance difference between point C and points A and B. 。

As shown in the figure, take the straight line L as the symmetry axis, and make the connecting line of points A' and A'b with respect to the straight line L intersect with point C, then point C is what you want. Reason: Find any point C ′ (different from point C) on the straight line L and connect CA, C ′ a, C ′ a ′, C ′ B. Because points A and A ′ are symmetrical with respect to the straight line L, Ca-C'A = C'A' ′-CB = A ′ B. And because point C ′ is on the L, C ′ a = C ′. In △ a ′ BC ′, c ′ a-c ′ b = c ′ a ′-c ′ b < a ′ b, so c ′ a ′.

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