The key to solving the shortest path problem in junior high school mathematics is to learn to make a symmetrical point of a fixed point about the straight line where the moving point is located, or to deal with it with translation and expansion diagram. This is helpful for us to solve this kind of problem with half the effort.

1、 ? Theoretical basis: the shortest line segment between two points, the shortest vertical line segment, the symmetry of points about lines, the translation of line segments, and the development diagram of three-dimensional graphics. Examples in the textbook are "drinking horses", "bridge site selection" and "three-dimensional development map"

2. Knowledge points: "The line segment between two points is the shortest", "The vertical line segment is the shortest", "The point is symmetrical about the line" and "The translation of the line segment". "drinking horses" and "building bridges and selecting sites" Often take an examination of "drinking horses", the background variants of the questions are angles, triangles, diamonds, rectangles, squares, trapezoid, circles, coordinate axes, parabolas and so on.

3. General idea of solving the problem: Find some symmetrical points about the line to realize "folding" to "straight". In recent years, there have been variations such as "three-fold line" to "straight".

The following are the solutions and key points of the corresponding problems. I hope you can practice more and attach the answers at last.