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贵刊85年第4期载有这么一道习题: △ABC中,∠A=45°,高AD分BC成BD=3,DC=2。求△ABC的面积。原文先后用几何法,三角法求解。这里再介绍一种解法,过程更为简捷,能为初中学生掌握。解设AD=x则AB=(9+x~2)~(1/2) (图右),AC=(4+x~2)~(1/2) 由面积公式得 S_(△ABC)=(1/2)AB·ACsinA =(1/2)BC·AD 用数值代换后化简得 x~4-37x~2+36=0 解之得 x_1~2=36,x_2~2=1(舍去) 于是 S_(△ABC)=(1/2)(45)~(1/2)·(40)~(1/2)/2~(1/2)·2/2=15 此法用面积公式布列方程,称作面积法,它在几何问题中的应用相当广泛。如第三届AIME试题中有一题是: 在一个面积为1的正方形中构作一个小正方形如下:将单位正方形的每一条边作n等分,然后如图所示将每个顶点与它相对的顶点最接
The 4th issue of the 85th edition of your journal contains such a problem: In ABC, ∠A=45°, and the high AD score is BC=BD=3 and DC=2. Find the area of △ABC. The original text was solved by geometric method and trigonometric method. Here again introduce a solution, the process is more simple and can be mastered for junior high school students. Solve for AD=x then AB=(9+x~2)~(1/2) (right), AC=(4+x~2)~(1/2) by the area formula S_(△ABC) = (1/2) AB · ACsinA = (1/2) BC · AD Substituting with a numerical value to obtain x~4-37x~2+36=0 Solution x_1~2=36, x_2~2= 1 (Third) Then S_(ΔABC)=(1/2)(45)~(1/2)·(40)~(1/2)/2~(1/2)·2/2=15 This method uses area equations to formulate a series of equations called area methods, which are widely used in geometric problems. For example, in the third AIME test question, a small square is constructed in a square with an area of 1 as follows: Each side of the unit square is n-divided, and then each vertex is opposed to it as shown in the figure. The vertices of the most connected