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1、中考数学专题复习研讨动点问题探究 等腰三角形分类讨论问题 新郑市实验中学 王淑敏 动点问题探究 等腰三角形分类讨论问题 图形中的点、线的运动，构成了数学中的一个新问题动态问题。 它通常分为三种类型:动点问题、动线问题、动形问题。 题型特点:此类问题常集代数、几何知识于一体，数形结合，有很强的综合性。是河南中招的必考题，且每年都为压轴题，以函数与三角形和四边形结合的题目为主。如08年为一次函数与三角形相结合，09年为二次函数与等腰三角形相结合，10年为二次函数与平行四边形相结合。 学情分析: 1、这类问题无论教师做了多大的努力,对学生来说都比较困难,所以一部分学生放弃作答。 2、一部分学生对动点
2、问题从根本上不理解,勉强照猫画虎,写了不少但不得分。 3、学生对动点问题有一定认识,对分类能进行简单尝试, 但不完整。 教学方法: 1、教师在教学时引导学生把动态问题变为静态问题来解,抓住变化中的“不变量” 。并从特殊位置点着手确定自变量取值范围, 对基本图形进行充分的分析,画出符合条件的各种草图分散难点、降低难度，将复杂问题简单化。 2、专题化，少而精。如动点问题有等腰三角形、直角三角形、三角形相似、 四边形存在性等问题，这些都需分类讨论，分小专题复习效果更好。 本节课重点来探究动态几何中的第一类型:动点问题等腰三角形分类讨论问题 ,一,自主解决,设计意图:为重点研讨作下铺垫, 1、在平面直
3、角坐标系中，已知点P(,2，,1).点T(t，0)是x轴上的一个动点。当t取何值时，?TOP是等腰三角形, 情况一:OP=OT T(,5,0);T(5,0)12情况二:PO=PT T3(-4,0) 5情况三:TO=TP T(,0)4 4设计意图:引导学生总结以已知线段为边作等腰三角形时通常要分三种情况讨论:以已知线段为底或为腰。且以已知线段为腰时以该腰不同顶点为顶角顶点有两种情况。 2、如图:已知平行四边形ABCD中，AB=7，BC=4，?A=30? (1)点P从点A沿AB边向点B运动，速度为1cm/s. 若设运动时间为t(s)，连接PC,当t为何值时，?PBC为等腰三角形, 若?PBC为等腰
4、三角形 则PB=BC ?t=3 ,二,师生互动探究新知 如图:已知平行四边形ABCD中，AB=7，BC=4，?A=30? (2)若点P从点A沿 射线AB运动，速度仍是1cm/s. 当t为何值时，?PBC为等腰三角形, ,小组合作交流讨论根据分类的标准易得到下面四种情况, 三、 4343或 时 ?PBC为等腰三角形 ?t=3或11或7+3设计意图:总结探究动点关键“化动为静分类讨论画出符合条件的各种草图” 注意一定要分开画. (三, 动脑创新再探新知:,两个动点问题 , ABCDADBCADDCABB?，?(,:354245如图，在梯形中，动点BCCNMB从点出发沿线段以每秒2个单位长度的速度向
5、终点运动;动点同CCDD时从点出发沿线段以每秒1个单位长度的速度向终点运动(设运动的时BCMNAB?ttA D 间为秒(1)求的长(2)当时，求的值( ?MNCt(3)试探究:为何值时，为等腰三角形( N B C M ,小组合作交流讨论, 分析:(1)如图? ，求出,=10 50?MNCGDCt,(2)由 求出 17解决动点问题的好助手: 数形结合定相似比例线段构方程 CNtCMt,，(102(3)当M、N运动到,秒时， NC为等腰三角形，须分三种情况讨论: 若?M10NCMC,tt,102?当时，即? t,3MNNC,NNEMC,?当时，过作于由等腰三角形三线合一性质得E11 ECMCtt,
6、1025，22ECt5,Rt?CEN在中， cosc,NCt53,t25CH3Rt?DHC,又在中，?解得t, cosc,t58CD511MNMC,MFCN,?当时，过作于F点. MFCNCt,22A D 1t60FC3N 2t,解得 cosC,F 17MCt1025,B C H M 256010?MNCt,t,综上所述，当t,、或时，为等腰三角形 8173总结:直角三角形能用相似解决的问题都能用三角函数法且用三角函数法针对性更强更省时间。 ,四,实践新知 提炼运用 在矩形ABCD中，AB,3cm，BC,4cm。设P，Q分别为BD，BC上的动点，在点P自点D沿DB方向作匀速运动的同时，点Q自点
7、B沿BC方向向点C作匀速运动，移动速度均为1cm/s,设点P，Q移动的时间为t(0t?4)。 (1)、写出PBQ的面积S(cm2)与时间t(s)之间的函数表达式，当t为何值时，S有最大值,最大值是多少, (2)、当t为何值时，PBQ为等腰三角形, (3)、PBQ能否成为等边三角形,若能，求t的值，若不能，说明理由, A D P B C M Q 331525 t，t(答案:1.S=,当t=时，s最大值= 10282 54025 2.t=, 213133 33.不能，tan60?=与tan?DBC=矛盾) 4 ,五,拓展延伸 体验中考 (09河南)如图，在平面直角坐标系中，已知矩形ABCD的三个顶
8、点 2B(4，0)、C(8，0)、D(8，8).抛物线y=ax+bx过A、C两点. (1)直接写出点A的坐标，并求出抛物线的解析式; (2)动点P从点A出发(沿线段AB向终点B运动，同时点Q从点C出发，沿线段CD向终点D运动(速度均为每秒1个单位长度，运动时间为t秒.过点P作PE?AB交AC于点E. ?过点E作EF?AD于点F，交抛物线于点G.当t为何值时，线段EG最长? ?连接EQ(在点P、Q运动的过程中，判断有几个时刻使得?CEQ是等腰三角形?请直接写出相应的t值. 分析:此题综合性更强，给学生充分的思考讨论时间。 尤其(2) ? 先求出EG与x的关系式，再求出EG最长时的x值，进而求出P
9、E的长，再由?APE? ?ABC或tan?BAC求,值. ?先由相似求出,与,的关系式，再分三种情况讨论. 12y,x，4x参考答案(1)A(4,8) 2(2)?,=4 5?,=40-16 40,= 1316,= 3课堂小结:让学生用自己的语言叙述，老师肯定正确的，纠正不准确的，并强调本节课重点. 1、化动为静，作出符合条件的各种情况的草图 2、分类讨论 3、数形结合 4、用三角形相似或三角函数法或勾股定理建立等量关系 化动为静 分类讨论 数形结合 思路 用三角形相似或三角函数法或勾股定理建立等量关系 (英文版 ) Two regulations promulgated for impleme
10、ntation 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-party supervision of major initiatives. The two regulations s
11、upporting 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 negative list, focusing on vertical gauge, draw the party organiz
12、ations 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 part is the interpretation of the two fundamental principles of t
13、he 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 of the party. and revised the necessity and revised history o
14、f 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 work lasted a Years, pooling the wisdom of the whole party, ideol
15、ogical 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 solve the masses of the people reflect a focus on the problem.
16、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 important speech, reflects the partys eighteen years comprehensive str
17、ictly 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 the party to manage the party strictly administering the deep
18、ening, 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 lack of prohibited provisions excessive, no autonomy requirements
19、; 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 existing problems, and is narrow, only needle of county-level leading
20、 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 have been unable to fully meet the comprehensive strictly admini
21、stering 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 the party constitution, damage the authority of Party Constit
22、ution 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 relevant regulations first amendment. By revising, really put the aut
23、hority 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 Committee of the Communist Party of China attaches great import
24、ance 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 the central deployment, the Central Commission for Discipline
25、 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 regulations revision, amendment clarifies the direction, major is
26、sues 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 Department The first party committee is mainly responsible for
27、 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 issued a notice to solicit the provinces (autonomous regions,
28、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 opinion. Central Commission for Discipline Inspection of extensive
29、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 of the two regulations revised draft. On October 18, the Cent
30、ral 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 regulations revision work and implement the partys eighteen, ten eight
31、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-t
32、erm, advance as a whole, with Bu Xiuding independent and . Main principle is: first, adhere to the party constitution to follow. The constitution about discipline and self-discipline required specific, awaken the party constitution of party compasses party consciousness, maintaining the authority of
33、 the constitution. General Secretary Xi Jinping pointed out that no rules, no side round. Party constitution is the fundamental law, the party must follow the general rules. In early 2015 held the eighth session of the Central Commission for Discipline Inspection Fifth Plenary Session of the 16th Ce
34、ntral Committee, Xi Jinping again pointed out that constitution is the party must follow the general rules, but also the general rules. the revision of the and is Method in adhere to the regulations established for the purpose of combining rule of virtue is to adhere to the party constitution as a f
35、undamental to follow, the constitution authority set up, wake up the party constitution and party rules the sense of discipline, the party constitution about discipline and self-discipline specific requirements. 4 second is to adhere to in accordance with the regulations governing the party and the
36、party. The Party of rule of virtue de, mainly refers to the partys ideals and beliefs, excellent traditional style. The revised the closely linked to the self-discipline, insisting on the positive initiative, for all members, highlight the vital few, emphasized self-discipline, focusing on the moral
37、ity, and the majority of Party members and the ideological and moral standards. The revised Ji method separately, Ji, Ji Yan to Method, as a negative list, emphasizing the heteronomy, focusing on vertical gauge. Is this one high and one low, a positive reaction, the strict party discipline and pract
38、ice results transformation for the integration of the whole party to observe moral and discipline requirements, for the majority of Party members and cadres provides benchmarking and ruler. Third, insist on to. In view of the problems existing in the party at the present stage, the main problems of
39、Party members and cadres in the aspect of self-discipline and abide by the discipline to make clearly defined, especially the partys eighteen years strict political discipline and political rules, organization and discipline and to implement the central eight provisions of the spirit against the fou
40、r winds and other requirements into Disciplinary provisions. Not one pace reachs the designated position, focusing on in line with reality, pragmatic and effective. After the revision of major changes, major changes in the and modified and needs to grasp several key problems (a) adhere to according
41、to regulations governing the party and party with morals in combination, for at the present stage, the leadership of the party members and cadres and Party members in existing main problems of self-discipline, put forward principles, requirements and specifications, showing Communists noble moral pu
42、rsuit, reflected at all times and in all over the world ethics from high from low 5 common requirements. One is closely linked to the self-discipline, removal and no direct relation to the provisions of . the second is adhere to a positive advocate, eight prohibition 52 are not allowed to about the
43、content of the negative list moved into synchronization amendment . Three is for all the party members, will apply object from the leadership of the party members and cadres to expand to all Party members, fully embodies the comprehensive strictly required. The fourth is prominent key minority, seiz
44、e the leadership of the party members and cadres is the key, and put forward higher requirements than the ordinary Party members. Five is to simplify, and strive to achieve concise, easy to understand, easy to remember. The revised is the ruling Party since the first insists on a positive advocate f
45、orAll Party members and the self-discipline norms, moral declaration issued to all members of the party and the National Peoples solemn commitment. criterion of a clean and honest administration consists of 4 parts, 18, more than 3600 words. After the revision of the , a total of eight, 281 words, i
46、ncluding lead, specification and Party member cadre clean fingered self-discipline norms, etc. Part 3 members low-cost clean and self-discipline, the main contents can be summarized as four must eight code. Lead part, reiterated on ideal and faith, fundamental purpose, the fine traditions and work s
47、tyle, noble sentiments, such as four must the principle of requirements, strong tone of self-discipline, The higher request for 6 and supervised tenet, the foothold in permanent Bao the partys advanced nature and purity, to reflect the revised standards requirements. Members of self-discipline norms around the party members how to corre