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运动与变化是解决数学问题的基本思想方法。数学中的许多概念,如函数、轨迹;许多方法,如换元、变形都体现了运动与变化的思想。在解题中,如果运用这种方法,有时能帮助我们确定解题的思路,下面以一道中考题为例说明之。例:已知矩形 ABCD 中,AB=1,点 M 在对角线AC 上,AM=1/4AC,直线 l 过点 M 且与 AC 垂直,与边 AD 相交于点 E。(1)如果 AD=3~(1/2),求证:点 B 在直线 l 上。(2)如果直线 l 与边 BC 相交于点 H(如图1),直线 l 把矩形分成两部分的面积之比为2:7,求 AD 的长。(3)如果直线 l 分别与边 AD、AB 交于点 E、G,
Movement and change are the basic ideas for solving math problems. Many concepts in mathematics, such as functions and trajectories; many methods, such as substitutions and deformations, embody the idea of movement and change. In the problem-solving problem, if this method is used, it can sometimes help us to determine the idea of problem solving. The following is an example of a mid-exam problem. Example: In the known rectangle ABCD, AB=1, point M is on the diagonal AC, AM=1/4AC, the straight line l crosses the point M and is perpendicular to the AC, and intersects the edge AD at the point E. (1) If AD=3~(1/2), verify that point B is on line l. (2) If the line l intersects the edge BC at point H (Figure 1), the ratio of the area of the line l to the rectangle divided into two parts is 2:7, and the length of AD is calculated. (3) If the straight line l intersects the sides AD, AB at points E, G, respectively,