浅谈勾股定理在初中教学中的应用 论 文.docx

1、浅谈勾股定理在初中教学中的应用 论 文浅谈勾股定理在初中教学中的应用 五德中学 曾 朋摘要：勾股定理是初中数学中非常重要的定理之一，他不仅是解直角三角形的重要依据，还揭示了直角三角形三边的关系，也体现了数形结合的思想，而且在初中数学教学中广泛应用。关键词：初中数学 勾股定理 应用一、勾股定理的历史背景勾股定理是“人类最伟大的十个科学发现之一”，是初等几何中的一个基本定理。这个定理有十分悠久的历史，几乎所有文明古国（希腊、中国、埃及、巴比伦、印度等）对此定理都有所研究。勾股定理在西方被称为毕达哥拉斯定理，相传是古希腊数学家兼哲学家毕达哥拉斯于公元前550年首先发现的。但毕达哥拉斯对勾股定理的证明

2、方法已经失传。著名的希腊数学家欧几里得在巨著几何原本（第卷，命题47）中给出一个很好的证明。中国古代对这一数学定理的发现和应用，远比毕达哥拉斯早得多。中国最早的一部数学著作周髀算经，在稍后一点的是九章算术。书中的勾股章说；“把勾和股分别自乘，然后把它们的积加起来，再进行开方，便可以得到弦”。 中国古代的数学家们不仅很早就发现并应用勾股定理，而且很早就尝试对勾股定理作理论的证明。最早对勾股定理进行证明的，是三国时期吴国的数学家赵爽。中国古代数学家们对于勾股定理的发现和证明，在世界数学史上具有独特的贡献和地位。尤其是其中体现出来的“数形统一”的思想方法，更具有科学创新的重大意义。二、勾股定理的证明

3、做四个全等的直角三角形，设它们的两条直角边长分别为a、b ，斜边长为c. 把它们拼成如图那样的一个多边形，使D、E、F在一条直线上. 过C作AC的延长线交DF于点P. D、E、F在一条直线上, 且RtGEF RtEBD, EGF = BED， EGF + GEF = 90， BED + GEF = 90， BEG =18090= 90.又 AB = BE = EG = GA = c， ABEG是一个边长为c的正方形. ABC + CBE = 90. RtABC RtEBD, ABC = EBD. EBD + CBE = 90. 即 CBD= 90.又 BDE = 90，BCP = 90，BC

4、= BD = a. BDPC是一个边长为a的正方形.同理，HPFG是一个边长为b的正方形.设多边形GHCBE的面积为S，则, 三、勾股定理的应用例题1：如图（8），水池中离岸边D点1.5米的C处，直立长着一根芦苇，出水部分BC的长是0.5米，把芦苇拉到岸边，它的顶端B恰好落到D点，并求水池的深度AC.解析：先将实物模型转化为数学模型，如图2. 由题意可知ACD中,ACD=90,在RtACD中，只知道CD=1.5，这是典型的利用勾股定理“知二求一”的类型。解：如图2，根据勾股定理，AC2+CD2=AD2 设水深AC= x米，那么AD=AB=AC+CB=xx22=（ x+0.5）2解之得x=2.

5、故水深为2米例题2： 如图3，正方形ABCD中，E是BC边上的中点，F是AB上一点，且那么DEF是直角三角形吗？为什么？解析：这道题把很多条件都隐藏了，乍一看有点摸不着头脑。仔细读题会意可以发现规律，没有任何条件，我们也可以开创条件，由可以设AB=4a，那么BE=CE=2 a,AF=3 a,BF= a,那么在RtAFD 、RtBEF和 RtCDE中，分别利用勾股定理求出DF,EF和DE的长，反过来再利用勾股定理逆定理去判断DEF是否是直角三角形。 解：设正方形ABCD的边长为4a,则BE=CE=2 a,AF=3 a,BF= a在RtCDE中，DE2=CD2+CE2=(4a)2+(2 a)2=2

6、0 a2同理EF2=5a2, DF2=25a2在DEF中，EF2+ DE2=5a2+ 20a2=25a2=DF2DEF是直角三角形，且DEF=90.例题3： 如图5，王师傅想要检测桌子的表面AD边是否垂直与AB边和CD边，他测得AD=80cm，AB=60cm，BD=100cm，AD边与AB边垂直吗？怎样去验证AD边与CD边是否垂直？解析：由于实物一般比较大，长度不容易用直尺来方便测量。我们通常截取部分长度来验证。矩形ABCD表示桌面形状，在AB上截取AM=12cm,在AD上截取AN=9cm(想想为什么要设为这两个长度？)，连结MN，测量MN的长度。1果MN=15,则AM2+AN2=MN2,所以

7、AD边与AB边垂直；2果MN=a15,则92+122=81+144=225, a2225,即92+122 a2，所以A不是直角。例4： 如图2-21所示已知：在正方形ABCD中，BAC的平分线交BC于E，作EFAC于F，作FGAB于G求证：AB2=2FG2分析 注意到正方形的特性CAB=45，所以AGF是等腰直角三角形，从而有AF2=2FG2，因而应有AF=AB，这启发我们去证明ABEAFE证 因为AE是FAB的平分线，EFAF，又AE是AFE与ABE的公共边，所以RtAFERtABE(AAS)，所以 AF=AB 在RtAGF中，因为FAG=45，所以AG=FG， AF2=AG2+FG2=2F

8、G2 由，得： AB2=2FG2例5: 如图2-23所示求证：任意四边形四条边的平方和等于对角线的平方和加对角线中点连线平方的4倍分析 如图2-23所示对角线中点连线PQ，可看作BDQ的中线证 设四边形ABCD对角线AC，BD中点分别是Q，P在BDQ中，即2BQ2+2DQ2=4PQ2+BD2 在ABC中，BQ是AC边上的中线，所以在ACD中，QD是AC边上的中线，所以将，代入得=4PQ2+BD2，即AB2+BC2+CD2+DA2=AC2+BD2+4PQ2总之，勾股定理在初中数学中单独命题较少，常与函数，圆，四边形、方程的知识综合在一起进行考查，在初中教学中应用广泛，是许多知识的桥梁，因此在初中

9、数学教学中应重视勾股定理的应用。（英文版 ） Two regulations promulgated for implementation is in the party in power for a long time and the rule of law conditions, the implementation of comprehensive strictly strategic plan, implementation in accordance with the rules and discipline to manage the party, strengthen inner

10、-party supervision of major initiatives. The two regulations supporting each other, the adhere to a positive advocate, focusing on morality is of Party members and Party leading cadres can see, enough to get a high standard; around the party discipline, disciplinary ruler requirements, listed as neg

11、ative list, focusing on vertical gauge, draw the party organizations and Party members do not touch the bottom line . Here, the main from four square face two party rules of interpretation: the first part introduces two party Revised regulations the necessity and the revision process; the second par

12、t is the interpretation of the two fundamental principles of the revision of laws and regulations in the party; the third part introduces two party regulations modified the main changes and needs to grasp several key problems; the fourth part on how to grasp the implementation of the two regulations

13、 of the party. and revised the necessity and revised history of the CPC Central Committee the amendment to the Chinese Communist Party members and leading cadres honest politics several guidelines and column 1 by 2015 to strengthen party laws and regulations focus. Two party regulations revision wor

14、k lasted a Years, pooling the wisdom of the whole party, ideological consensus, draw historical experience, respect for the wisdom of our predecessors, which reflects the unity of inheritance and innovation; follow the correct direction, grasp the limited goals, adhere to the partys leadership, to s

15、olve the masses of the people reflect a focus on the problem. The new revision of the and , reflects the partys 18 and the eighth session of the third, the spirit of the fourth plenary session, reflecting the experience of studying and implementing the General Secretary Xi Jinping series of importan

16、t speech, reflects the partys eighteen years comprehensive strictly practice. (a) revised two regulations of the party need of in based on revised, the promulgation and implementation of January 2010, to strengthen the construction of the contingent of leading cadres play an important role. But with

17、 the party to manage the party strictly administering the deepening, has not been able to fully meet the actual needs. Content is too complicated, eight prohibition, 52 are not allowed to hard to remember, and also difficult to put into practice; the second is concisely positive advocated by the lac

18、k of prohibited provisions excessive, no autonomy requirements; the third is banned terms and discipline law, both with the party discipline, disciplinary regulations repeat and Criminal law and other laws and regulations repeat; the fourth is to clean the theme is not prominent, not for the existin

19、g problems, and is narrow, only needle of county-level leading cadres above. is in 1997 based on revision, in December 2003 the promulgation and implementation, to strengthen the construction of the party play very important role. Along with the development of the situation, which many provisions ha

20、ve been unable to fully meet the comprehensive strictly administering the practice needs. One is Ji law, more than half of the provisions and criminal law and other countries laws and regulations Repetition; two is the political discipline regulations is not prominent, not specific, for violation of

21、 the party constitution, damage the authority of Party Constitution of misconduct lack necessary and serious responsibility to pursue; third is the main discipline for the leading cadres, does not cover all Party members. Based on the above situation, need to and the two is likely to be more relevan

22、t regulations first amendment. By revising, really put the authority of Party discipline, the seriousness in the party tree and call up the majority of Party members and cadres of the party constitution of party compasses party consciousness. (II) two party regulations revision process the Central C

23、ommittee of the Communist Party of China attaches great importance to two regulations revision . Xi Jinping, general books recorded in the Fifth Plenary Session of the eighth session of the Central Commission for Discipline Inspection, on the revised regulations made clear instructions. According to

24、 the central deployment, the Central Commission for Discipline Inspection from 2014 under six months begin study two regulations revision. The Standing Committee of the Central Commission for Discipline Inspection 4 review revised. Comrade Wang Qishan 14 times held a special meeting to study two reg

25、ulations revision, amendment clarifies the direction, major issues of principle, path and target, respectively held a forum will listen to part of the province (area) secretary of the Party committee, Secretary of the Discipline Inspection Commission, part of the central ministries and state organs

26、Department The first party committee is mainly responsible for people, views of experts and scholars and grassroots party organizations and Party members. Approved by the Central Committee of the Communist Party of China, on 7 September 2015, the general office of the Central Committee of the Party

27、issued a notice to solicit the provinces (autonomous regions, municipalities) Party, the central ministries and commissions, state ministries and commissions of the Party (party), the General Political Department of the military, every 3 people organization of Party of two regulations revision opini

28、on. Central Commission for Discipline Inspection of extensive solicitation of opinions, careful study, attracting, formed a revised sent reviewers. In October 8 and October 12, Central Committee Political Bureau Standing Committee and the Political Bureau of the Central Committee After consideration

29、 of the two regulations revised draft. On October 18, the Central Committee of the Communist Party of China formally issued two regulations. Can say, two laws amendment concentrated the wisdom of the whole party, embodies the party. Second, and revision of the basic principles of two party regulatio

30、ns revision work and implement the partys eighteen, ten eight plenary, the spirit of the Fourth Plenary Session of the Eleventh Central Committee and General Secretary Xi Jinping important instructions on the revised and , highlighting the ruling party characteristics, serious discipline, the discipline quite in front of the law, based on the current, a long-term, advance as a whole, with Bu Xiuding independent and . Main principle is: first, adhere to the party constitution to follow.