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勾股定理是平面几何中的重要定理,其应用极其广泛,在应用勾股定理时,应注意以下几点:一、要注意正确使用勾股定理例1:在 Rt△ABC 中,∠B=Rt∠,a=1,b=3~(1/2),求 C。错解:根据勾股定理,得 a~2+b~2=c~2,从而有 c=(a~2+b~2)~(1/2)=((1+3~(1/2))~2)~(1/2)=2。分析:上述解答错误的原因是应用勾股定理时,只注意表面形式,只有当∠C=Rt∠时,勾股定理的表达式应为 a~2+b~2=c~2,而当∠B=Rt∠时,勾股定理的表达式应为 a~2+c~2=b~2,正确答案为:c=(b~2-a~2)~(1/2)=((3~(1/2))~2-1~2)~(1/2)=2~(1/2)。二、要注意定理存在的条件例2 在边长为整数的△ABC 中,AB>AC。如果 AC=4,BC=3,求 AB 的长。
The Pythagorean theorem is an important theorem in plane geometry. Its application is extremely wide. When applying the Pythagorean theorem, the following points should be noted: 1. Pay attention to the correct use of the Pythagorean theorem Example 1: In Rt △ ABC, ∠B= Rt∠, a=1, b=3~(1/2), find C. Misunderstanding: According to the Pythagorean theorem, we get a~2+b~2=c~2, so c=(a~2+b~2)~(1/2)=((1+3~(1/ 2))~2)~(1/2)=2. Analysis: The reason for the above-mentioned solution to the wrong answer is to apply the Pythagorean theorem, and only pay attention to the surface form. Only when ∠C=Rt∠, the expression of the Pythagorean theorem should be a~2+b~2=c~2, and when When ∠B=Rt∠, the expression of the Pythagorean theorem should be a~2+c~2=b~2. The correct answer is: c=(b~2-a~2)~(1/2)=( (3~(1/2))~2-1~2)~(1/2)=2~(1/2). Second, we must pay attention to the condition of the existence of the theorem 2. In the case of △ABC with an integer side length, AB>AC. If AC=4 and BC=3, find the length of AB.