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问题将边长为a的正方形纸ABCD的一个顶点A“折”到它的对边CD上。从操作过程发现“折”的过程其实就是数学中的“对称”。在《几何画板》软件中,可选取边CD上任意一点A_1,则线段AA_1的中垂线MN就是“折线”。如图1所示,拖动点A_1,观察图形动态的变化过程。可以发现在这一简单的过程中蕴含着丰富的量的变化和形的运动。从中提出一些数学问题进行探讨能发现一些有趣的结论。
The problem “folds” one of the vertices A of the square paper ABCD with a side length a onto its opposite CD. The process of finding “folding” from the operation is actually “symmetry” in mathematics. In the “Geometric Sketchpad” software, any point A_1 on the side CD can be selected, and the perpendicular line MN of the line segment AA_1 is a “polyline”. As shown in Fig. 1, drag the point A_1 to observe the dynamic change process of the graphic. It can be found that in this simple process there is a rich amount of change and shape movement. From the point of view of some mathematics issues, we can find some interesting conclusions.