初中数学比例线段

初中数学比例线段 一、授课目的: 1,理解和应用比例的性质、 2，掌握平行线分线段成比例定理，并能熟练应用 二、授课内容: 知识考点: 本节知识在历年中考的考题中，主要涉及用比例的性质、平行线分线段成比例定理。由于比例的性质在应用时有其限制条件，一些中考题又以此为背景分类求解题。 精典例题: x,y，zxyz【例1】已知,,,0，那么, 。 345x，y，z 分析:此类问题有多种解法，一是善于观察所求式子的特点，灵活运用等比性质求解;二是利用方程的观 43kx,zy,z点求解，将已知条件转化为，，代入所求式子即可得解;三是设“”值法求解，这种方法对55 于解有关连比的问题十分方便有效，要掌握好这一技巧。 1: 3 a,2c，e,2ace2变式1:已知，若，则, 。 ,,,b,2d，f,3,0bdf3b,2d，f,3 2x,y，3z变式2:已知，求的值。 x:y:z,2:1:3x，2y a，b,ca,b，cb，c,akk,,,变式3:已知，则的值为 。 cba 2答案:(1);(2)3;(3)1或,2; 3 【例2】如图，在?ABC中，点E、F分别在AB、AC上，且AE,AF，EF的延长线交BC的延长线于点D。求证:CD?BD,CF?BE。 分析:在题设中，没有平行的条件，要证明线段成比例，可考虑添加平行线，观察图形，对照结论，需要变换比CF?BE，为了变换比CF?BE，可以过点C作BE的平行线交ED于G，并设法证明CG,CF即可获证。 A AAE F EE FFBGGDC BDBDCCG 例2图1 例2图2 例2图3 risk managers with two account manager to deal with the above key. Section I of chapter III risk management requirements pre-loan investigation pre-loan investigation should double handle. Pre-loan investigation dominated by field investigations on the ground, while utilizing credit enquiries, telephone enquiries, bank information system query-related personnel, and so on. Interview system of Bank's strict application of the system of personal credit business interview. Investigators should work with borrowers, to the repayment or the guarantor credit business, such as parties and the spouse of confirmed acts such as loans, guarantees, verification of its authenticity and its rights and obligations in the credit business, retain these signed transcript of the interview, from the first step to prevent false loans and other business risks. Investigator should retain the borrower sign a personal loan application, loan applications should be clearly the amount, purpose, duration and other information. Investigators should follow the practice for information collection and verification of data authenticity it carefully, and keep copies, photocopies must be checked against the original agreement, effectively preventing borrowers using false information to apply for credit. Income and occupation, and other key information to verify authenticity investigators should carefully verifying the borrower's income. To income as the main source of repayment, subject to proof of income is required, but should also audit payroll flowing water, and staff than the peer confirmed the authenticity of the same type; operating income as the main source of repayment, and should refer to the working capital loans required to verify real incomes, accuracy; guard no ability to repay or guarantee of the ability of the borrower, guarantor to obtain credit. Given that some products for an object to have a specific professional restrictions 本例为了实现将比CF?BE转换成比CD?BD的目的，还有多种不同的添画平行线的方法，它们的共同特征都是构造平行线截得的线段成比例的基本图形，请你们参考图形，自己去构思证明。 AEA GAEFFEF CBDBCDBCD 例2图4 变式1图 变式2图 AFAE1变式1:已知如图，D是?ABC的边BC的中点，且，求的值。 ,BE3FC 变式2:如图，BD?DC,5?3，E为AD的中点，求BE?EF的值。 1答案:(1);(2)13?3; 3 【例3】如图，在?ABC中，P为中线AM上任一点，CP的延长线交AB于D，BP的延长线交AC于E，连结DE。 (1)求证:DE?BC; (2)如图，在?ABC中，DE?BC，DC、BE交于P，连结AP并延长交BC于M，试问:M是否为BC的中点, 解析:(1)延长AM至Q，使MQ,MP ?BM,MC，?四边形BPCQ是平行四边形 A ?CD?BQ，BE?QC DEADAPAE ? ,,PDBPQEC MCB ?DE?BC (2)过B作BQ?CD交AM的延长线于Q QADAPAE例3图 ?DE?BC，?,, DBPQEC APAE ?，?BE?QC ,PQEC ?四边形BPCQ是平行四边形 ?M是BC的中点 探索与创新: 【问题】请阅读下面，并回答所提出的问题: 三角形内角平分线性质定理:三角形的内角平分线分对边所得的两条线段和这个角的两边对应成比例。如 BDAB,图，?ABC中，AD是角平分线。求证:。 DCAC BDAB,分析:要证，一般只要证BD、DC与AB、AC或BD、AB与DC、AC所在三角形相似，现在DCAC BDAB,B、D、C在同一条直线上，?ABD与?ADC不相似，需要考虑用别的方法换比。我们注意到在比例式DCACrisk managers with two account manager to deal with the above key. Section I of chapter III risk management requirements pre-loan investigation pre-loan investigation should double handle. Pre-loan investigation dominated by field investigations on the ground, while utilizing credit enquiries, telephone enquiries, bank information system query-related personnel, and so on. Interview system of Bank's strict application of the system of personal credit business interview. Investigators should work with borrowers, to the repayment or the guarantor credit business, such as parties and the spouse of confirmed acts such as loans, guarantees, verification of its authenticity and its rights and obligations in the credit business, retain these signed transcript of the interview, from the first step to prevent false loans and other business risks. Investigator should retain the borrower sign a personal loan application, loan applications should be clearly the amount, purpose, duration and other information. Investigators should follow the practice for information collection and verification of data authenticity it carefully, and keep copies, photocopies must be checked against the original agreement, effectively preventing borrowers using false information to apply for credit. Income and occupation, and other key information to verify authenticity investigators should carefully verifying the borrower's income. To income as the main source of repayment, subject to proof of income is required, but should also audit payroll flowing water, and staff than the peer confirmed the authenticity of the same type; operating income as the main source of repayment, and should refer to the working capital loans required to verify real incomes, accuracy; guard no ability to repay or guarantee of the ability of the borrower, guarantor to obtain credit. Given that some products for an object to have a specific professional restrictions 中，AC恰好是BD、DC、AB的第四比例项，所以考虑过C作CE?AD交BA的延长线于E，从而得到BD、 BDABCD、AB的第四比例项AE，这样，证明就可以转化为证AE,AC。 ,DCAC 证明:过C作CE?AD交BA的延长线于E ，1,，2,,E,,,，2,，3 CE?AD?E,?3 ,,,A,,，1,，E,,12 AE,AC ,3 BDABDCB, CE?AD ,DCAE问题图 BDAB, ? DCAC (1)上述证明过程中，用了哪些定理(写出两个定理即可); (2)在上述分析、证明过程中，主要用到了三种数学思想的哪一种,选出一个填入后面的括号内( ) ?数形结合思想 ?转化思想 ?分类讨论思想 答案:?转化思想 (3)用三角形内角平分线性质定理解答问题:已知AD是?ABC中?BAC的角平分线，AB,5 cm，AC,4 cm，BC,7 cm，求BD的长。 35答案:cm 9 评注:本题的目的主要在于考查学生的阅读理解能力。 跟踪训练: 一、填空题: 2m,n1m,1、若，则, ;若，且，则 , ，y, ，xx:y:z,2:4:73x,y，2z,32n3n , 。 z x，yy，zx，zk2、若，则, 。 ,,,kzxy 3、已知数3、6，请再写出一个数，使这三个数中的一个数是另外两个数的比例中项，则这个数是 。 4、如图，在?ABCD中，E为BC上一点，BE?EC,2?3，AE交BD于点F，则BF?FD, 。 二、选择题: 1、已知如图，AB?CD，AD与BC相交于点O，则下列比例式中正确的是( ) BCOBABOAOAOBABOB,,,, A、 B、 C、 D、 CDADODBCCDOCADOD A ABADEDO GFF ECDBCCB 选择第1题图 填空第4题图 选择第2题图 2、如图，在?ABC中，AD,DF,FB，AE,EG,GC，FG,4，则( ) risk managers with two account manager to deal with the above key. Section I of chapter III risk management requirements pre-loan investigation pre-loan investigation should double handle. Pre-loan investigation dominated by field investigations on the ground, while utilizing credit enquiries, telephone enquiries, bank information system query-related personnel, and so on. Interview system of Bank's strict application of the system of personal credit business interview. Investigators should work with borrowers, to the repayment or the guarantor credit business, such as parties and the spouse of confirmed acts such as loans, guarantees, verification of its authenticity and its rights and obligations in the credit business, retain these signed transcript of the interview, from the first step to prevent false loans and other business risks. Investigator should retain the borrower sign a personal loan application, loan applications should be clearly the amount, purpose, duration and other information. Investigators should follow the practice for information collection and verification of data authenticity it carefully, and keep copies, photocopies must be checked against the original agreement, effectively preventing borrowers using false information to apply for credit. Income and occupation, and other key information to verify authenticity investigators should carefully verifying the borrower's income. To income as the main source of repayment, subject to proof of income is required, but should also audit payroll flowing water, and staff than the peer confirmed the authenticity of the same type; operating income as the main source of repayment, and should refer to the working capital loans required to verify real incomes, accuracy; guard no ability to repay or guarantee of the ability of the borrower, guarantor to obtain credit. Given that some products for an object to have a specific professional restrictions A、DE,1，BC,7 B、DE,2，BC,6 C、DE,3，BC,5 D、DE,2，BC,8 3、如图，BD、CE是?ABC的中线，P、Q分别是BD、CE的中点，则PQ?BC,( ) A、1?3 B、1?4 C、1?5 D、1?6 2AE,kECk4、如图，?，，BC,4CD，若，则,( ) AF,FBll215 55A、 B、2 C、 D、4 32 AAGAl1 FDEDEKEQPl2DBCBCHCBF选择第3题图 选择第4题图 解答第1题图 三、解答题: DK1、已知如图，AD,DE,EC，且AB?DF?EH，AH交DF于K，求的值。 KF 2、如图，?ABCD中，EF交AB的延长线于E，交BC于M，交AC于P，交AD于N，交CD的延长 PE,PM,PF,PN线于F。求证:。 AFm,3、如图，在?ABC中，AC,BC，F为底边AB上一点，(、,0)，取CF的中点D，连结mnBFn AD，并延长交BC于E。 BE(1)求的值; EC (2)如果BE,2EC，那么CF所在的直线与边AB有怎样的位置关系,并证明你的结论; m(3)E点能否为BC的中点,如果能，求出相应的的值;如果不能，说明理由。 n 4、如图，已知梯形ABCD中，AD?BC，AB,DC,3，P为BC上一点，PE?AB交AC于E，PF?CD bx,a，b交BD于F，设PE、PF的长分别为、，。那么当点P在BC边上移动时，的值是否变化,若变ax化，求出的范围;若不变，求出的值，并说明理由。 xx EC MCB ADEDPEFDAN AFBFPCB 解答第2题图 解答第3题图 解答第4题图 跟踪训练参考答案 一、填空题: risk managers with two account manager to deal with the above key. Section I of chapter III risk management requirements pre-loan investigation pre-loan investigation should double handle. Pre-loan investigation dominated by field investigations on the ground, while utilizing credit enquiries, telephone enquiries, bank information system query-related personnel, and so on. Interview system of Bank's strict application of the system of personal credit business interview. Investigators should work with borrowers, to the repayment or the guarantor credit business, such as parties and the spouse of confirmed acts such as loans, guarantees, verification of its authenticity and its rights and obligations in the credit business, retain these signed transcript of the interview, from the first step to prevent false loans and other business risks. Investigator should retain the borrower sign a personal loan application, loan applications should be clearly the amount, purpose, duration and other information. Investigators should follow the practice for information collection and verification of data authenticity it carefully, and keep copies, photocopies must be checked against the original agreement, effectively preventing borrowers using false information to apply for credit. Income and occupation, and other key information to verify authenticity investigators should carefully verifying the borrower's income. To income as the main source of repayment, subject to proof of income is required, but should also audit payroll flowing water, and staff than the peer confirmed the authenticity of the same type; operating income as the main source of repayment, and should refer to the working capital loans required to verify real incomes, accuracy; guard no ability to repay or guarantee of the ability of the borrower, guarantor to obtain credit. Given that some products for an object to have a specific professional restrictions 231、，4，8，14;2、2或,1;3、或或12等;4、2?5; 3232二、选择题:CBBB 三、解答题: 11、; 3 PEPN,2、证明即可; PFPM BEm，n3、(1);(2)直线EF垂直平分AB;(3)E不能是BC的中点; ,ECn x,34、的值不变化，为定值，。 x risk managers with two account manager to deal with the above key. Section I of chapter III risk management requirements pre-loan investigation pre-loan investigation should double handle. Pre-loan investigation dominated by field investigations on the ground, while utilizing credit enquiries, telephone enquiries, bank information system query-related personnel, and so on. Interview system of Bank's strict application of the system of personal credit business interview. Investigators should work with borrowers, to the repayment or the guarantor credit business, such as parties and the spouse of confirmed acts such as loans, guarantees, verification of its authenticity and its rights and obligations in the credit business, retain these signed transcript of the interview, from the first step to prevent false loans and other business risks. Investigator should retain the borrower sign a personal loan application, loan applications should be clearly the amount, purpose, duration and other information. Investigators should follow the practice for information collection and verification of data authenticity it carefully, and keep copies, photocopies must be checked against the original agreement, effectively preventing borrowers using false information to apply for credit. Income and occupation, and other key information to verify authenticity investigators should carefully verifying the borrower's income. To income as the main source of repayment, subject to proof of income is required, but should also audit payroll flowing water, and staff than the peer confirmed the authenticity of the same type; operating income as the main source of repayment, and should refer to the working capital loans required to verify real incomes, accuracy; guard no ability to repay or guarantee of the ability of the borrower, guarantor to obtain credit. Given that some products for an object to have a specific professional restrictions