特殊平行四边形复习题

特殊的平行四边形 22、(本满分10分)如图，已知平行四边形ABCD，过A作AM?BC与M，交BD于E，过C作CN?,,于,，交,,于,，连结AF、CE. (1)求证:四边形AECF为平行四边形; (2)当AECF为菱形，M点为BC的中点时，求AB:AE的值。 考点:平行四边形的判定，菱形的判定 解答:(1)证明:因为AE?BC，所以?AMB=90?， 因为CN?,,，所以?CNA=90? 又因为BC?AD，所以?BCN=90? 所以AE?CF 又由平行得?ADE=?CBD,AD=BC 所以?ADE??BCF，所以AE=CF 因为AE?CF，AE=CF所以四边形AECF为平行四边形. (2)当平行四边形AECF为菱形时，连结AC交BF于点O， 则AC与EF互相垂直平分， 又OB=OD, 所以AC与BD互相垂直平分 所以，四边形ABCD为菱形 所以AB=BC 因为M是BC的中点，AM?BC， 所以?ABM??CAM， 所以AB=AC 为等边三角形， 所以?ABC=60?，?CBD=30? 3在RT?BCF中，CF:BC=tan?CBF=, 3 又AE=CF,AB=BC, 3所以AB:AE= 点评:本题考查了平行四边形的判定，菱形的性质，解直角三角形的有关知识。解决此类综合问题的关键在于根据已知图形，联想到它的性质，选择其中的部分性质进行计算或证明。 (2013重庆，24，10分)已知:如图，在菱形ABCD中，F为边BC的中点，DF与对角线AC交于点M，过M作ME?CD于点E,?1=?2。 (1)若CE=1，求BC的长;(2)求证AM=DF+ME。 within explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall meet the following requirements: repair and decoration works should end second grouting and finishing work should end, satisfactory operation of explosion-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-proof electrical equipment into the line port and wiring and cables, wires should be capable of reliably sealing, extra inlet elastic gasket, metal gasket should be complete and should tighten the clamping larva inlet seals. Metal gasket thickness shall not be less than 2mm. 4.1.11.4 the installation of explosion-proof lamps lamp type, model and power, the design and specification of requirements, may not be altered. Spiral bulbs should be tightened up and good contact, not loose, lamp housings should be complete, bolts should be tightened. 4.1.11.5 explosion dangerous environment of cable cannot directly connection, in non-normal situation Xia, must in corresponding of explosion-proof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories wall Shi, must take filling sand, and fill resistance fire blocking material or added set fire wall, protection tube ends of tube mouth at, should will cable around with non-burning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is less than pipe diameter, and shall not is less than 40mm. Introduction of explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-proof electric junction box, explosion-proof requirements should be to cope with different environmental conditions of different materials should be used to find the connection. 4.1.11.6 explosive AB1 FM 2 CDE 解析:延长DF,BA交于G，可证?CEM??CFM, ?CDF??BGF,通过线段的简单运算，即可求得。 答案:(1)?四边形ABCD是菱形?CB=CD,AB?CD??1=?ACD ,??1=?2 ??2=?ACD ?MC=MD ?ME?CD ?CD=2CE=2 ?BC=CD=2 (2) 延长DF,BA交于G，?四边形ABCD是菱形??BCA=?DCA , ?BC=2CF,CD=2CE ?CE=CF ?CM=CM??CEM??CFM, ?ME=MF?AB?CD??2=?G, ?GBF=?BCD?CF=BF??CDF??BGF?DF=GF??1=?2, ?G=?2??1=?G?AM=GM=MF+GF=DF+ME 点评:利用三角形全等来解决线段的有关问题是常见的思考方法，遇到中点延长一倍，是常见的辅助性做法。 AGB1 MF 2 CDE (2013山东省临沂市，22，7分)如图，点A、F、C、D在同一直线上，点B和点E分别在直线AD的两侧，且AB=DE,?A=?D,AF=DC. (1)求证:四边形BCEF是平行四边形; 0(2)若?ABC=90，AB=4，BC=3，当AF为何值时，四边形BCEF是菱形。 【解析】(1)证明?ABC??DEF，即可得到?BC=EF,BC?EF,根据一组对边平行且相等的四边形是平行四边形即可判断; 0(2)假设四边形BCEF是菱形，连接BE，当?ABC=90，AB=4，BC=3时，应用勾股定理可求 2222AB，BC,4，3,5得AC=,可求得?ABC??BGC,应用三角形相似的性质求得 plete, bolts should be tightened. 4.1.11.5 explosion dangerous environment of cable cannot directly connection, in nonequirements, may not be altered. Spiral bulbs should be tightened up and good contact, not loose, lamp housings should be comproof lamps lamp type, model and power, the design and specification of r-be less than 2mm. 4.1.11.4 the installation of explosion c gasket, metal gasket should be complete and should tighten the clamping larva inlet seals. Metal gasket thickness shall notelasti proof electrical equipment into the line port and wiring and cables, wires should be capable of reliably sealing, extra inlet-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-sfactory operation of explosionmeet the following requirements: repair and decoration works should end second grouting and finishing work should end, satiwithin explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall erials should be used to find the connection. 4.1.11.6 explosiveproof requirements should be to cope with different environmental conditions of different mat-proof electric junction box, explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-of explosionshall not is less than 40mm. Introduction burning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is less than pipe diameter, and-set fire wall, protection tube ends of tube mouth at, should will cable around with nonwall Shi, must take filling sand, and fill resistance fire blocking material or added proof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -tion Xia, must in corresponding of explosionnormal situa-2 77AF=，所以当AF=时，四边形BCEF是菱形( 55 解:(1)读图分析线段FC是公共部分，?AF=DC, ?AF+FC=DC+FC,即AC=DF， 又?AB=DE,?A=?D,??ABC??DEF, ?BC=EF,?ACB=?DFE. EF,?四边形BCEF是平行四边形; ?BC? (2)若四边形BCEF是菱形，连接BE，交CF于点G， ?BE?CF,FG=CG, 0??ABC=90，AB=4，BC=3，由勾股定理得， 2222AC=, AB，BC,4，3,5 0??BGC=?ABC=90，?ACB=?BCG, ??ABC??BGC, BCCG3CG918?,即,CG=,?FC=2CG=. ,,5553ACBC 187?AF=AC-FC=5-=. 55 7?当AF=时，四边形BCEF是菱形。 5 主要考查了平行四边形的判定以及菱形的判定方法，正确掌握判定定理是解【点评】本题 题的关键( 23.3 正方形 (2013贵州铜仁，18，4分以边长为2的正方形的中心O为端点，引两条相互垂直的射线， 分别与正方形的边交于A、B两点，则线段AB的最小值是__________( 【解析】如图?四边形CDEF是正方形， ??OCD=?ODB=45?，?COD=90?，OC=OD， ?AO?OB， ??AOB=90?， ??COA+?AOD=90?，?AOD+?DOB=90?， ??COA=?DOB， ?在?COA和?DOB中， ??COA??DOB， proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-s than pipe diameter, and shall not is less than 40mm. Introduction of explosionburning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is les-uld will cable around with none filling sand, and fill resistance fire blocking material or added set fire wall, protection tube ends of tube mouth at, showall Shi, must takproof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -normal situation Xia, must in corresponding of explosion-environment of cable cannot directly connection, in nonrous ghtened up and good contact, not loose, lamp housings should be complete, bolts should be tightened. 4.1.11.5 explosion dangelamps lamp type, model and power, the design and specification of requirements, may not be altered. Spiral bulbs should be ti proof-ten the clamping larva inlet seals. Metal gasket thickness shall not be less than 2mm. 4.1.11.4 the installation of explosionles, wires should be capable of reliably sealing, extra inlet elastic gasket, metal gasket should be complete and should tighproof electrical equipment into the line port and wiring and cab-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-works should end second grouting and finishing work should end, satisfactory operation of explosionmeet the following requirements: repair and decoration within explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall losivebe to cope with different environmental conditions of different materials should be used to find the connection. 4.1.11.6 exp proof requirements should-proof electric junction box, explosion-3 ?OA=OB， ??AOB=90?， ??AOB是等腰直角三角形， 由勾股定理得:AB==OA， 要使AB最小，只要OA取最小值即可， 根据垂线段最短，OA?CD时，OA最小， ?正方形CDEF， ?FC?CD，OD=OF， ?CA=DA， ?OA=CF=1， ?AB=OA= 2 【解答】. 【点评】本题考查了正方形的性质、垂线段最短、全等三角形的判定与性质、直角三角形斜边上的中线、等知识，题目具有代表性，有一定的难度。解答本题关键是判断AB=2OA时，AB最小，即OA与OB分别与正方形边长垂直时AB有最小值。 ( 2013年浙江省宁波市,12,3)勾股定理是几何中的一个重要定理，在我国古算书《周髀算经》中，就有“若勾三，股四，则弦五”记载，如图1是由边长相等的小正形和直角三角形构成的可以用其面积关系验证勾股定理。图2是由图1放入矩形内得到的，? 0BAC=90,AB=3,AC=4,D,E,F,G,H,I都在矩形KLMJ的边上，则矩形KLMJ的面积为 (A)90 (B)100 (C)110 (D)121 【解析】如图，延长AB交KF于点O，延长AC交GM于点P， 所以，四边形AOLP是正方形， 边长AO=AB+AC=3+4=7， 所以，KL=3+7=10，LM=4+7=11， 因此，矩形KLMJ的面积为10×11=110( 故选C( 【答案】C 【点评】本题考查了勾股定理的证明，作出辅助线构造出正方形是解题的关键( -tion Xia, must in corresponding of explosionnormal situa-plete, bolts should be tightened. 4.1.11.5 explosion dangerous environment of cable cannot directly connection, in nonequirements, may not be altered. Spiral bulbs should be tightened up and good contact, not loose, lamp housings should be comproof lamps lamp type, model and power, the design and specification of r-be less than 2mm. 4.1.11.4 the installation of explosion c gasket, metal gasket should be complete and should tighten the clamping larva inlet seals. Metal gasket thickness shall notelasti proof electrical equipment into the line port and wiring and cables, wires should be capable of reliably sealing, extra inlet-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-sfactory operation of explosionmeet the following requirements: repair and decoration works should end second grouting and finishing work should end, satiwithin explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall erials should be used to find the connection. 4.1.11.6 explosiveproof requirements should be to cope with different environmental conditions of different mat-proof electric junction box, explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-of explosionshall not is less than 40mm. Introduction burning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is less than pipe diameter, and-set fire wall, protection tube ends of tube mouth at, should will cable around with nonwall Shi, must take filling sand, and fill resistance fire blocking material or added proof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories 4 (2013四川内江，21，9分)如图11，四边形ABCD是矩形，E是BD上的一点，?BAE,?BCE，?AED,?CED，点G是BC、AE延长线的交点，AG与CD相交于点F( (1)求证:四边形ABCD是正方形; (2)当AE,2EF时，判断FG与EF有何数量关系,并证明你的结论( B A E C D F G 图11 )四边形ABCD是矩形，只需证得一组邻边相等即可说明它是正方形(接下【解析】(1 来通过证明?AED??CED得AD,CD解决问题((2)由(1)中全等三角形得AE,CE，?DAE,?DCE，再由BG?AD得?G,?EAD，从而?DCE,?G，这样就可证明?CEG??FEC，由它产生相似比并结合AE,2EF即可得解( 【答案】解:(1)证明:?四边形ABCD是矩形，??BAD,?BCD,90?( ??BAE,?BCE，??BAD,?BAE,?BCD,?BCE，即?EAD,?ECD( ??AED,?CED，ED,ED，??AED??CED(?AD,CD( ?矩形ABCD是正方形( (2)FG,3EF( 理由:?BG?AD，??G,?EAD( 由于?EAD,?ECD，??G,?ECD( CEEG??CEG,?FEC，??CEG??FEC(?,( EFCE 由(1)知CE,AE，而AE,2EF，故CE,2EF( environment of cable cannot directly connection, in nonrous ghtened up and good contact, not loose, lamp housings should be complete, bolts should be tightened. 4.1.11.5 explosion dangelamps lamp type, model and power, the design and specification of requirements, may not be altered. Spiral bulbs should be ti proof-ten the clamping larva inlet seals. Metal gasket thickness shall not be less than 2mm. 4.1.11.4 the installation of explosionles, wires should be capable of reliably sealing, extra inlet elastic gasket, metal gasket should be complete and should tighproof electrical equipment into the line port and wiring and cab-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-works should end second grouting and finishing work should end, satisfactory operation of explosionmeet the following requirements: repair and decoration within explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall losivebe to cope with different environmental conditions of different materials should be used to find the connection. 4.1.11.6 exp proof requirements should-proof electric junction box, explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-s than pipe diameter, and shall not is less than 40mm. Introduction of explosionburning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is les-uld will cable around with none filling sand, and fill resistance fire blocking material or added set fire wall, protection tube ends of tube mouth at, showall Shi, must takproof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -normal situation Xia, must in corresponding of explosion-5 ?EG,2CE,4EF，即EF，FG,4EF( ?FG,3EF( 【点评】本题综合考查了矩形、正方形、全等三角形、相似三角形知识，题目条件简洁明了，突出了对基础知识、核心知识的交叉考查，是一道中档好题(解决问题(2)，还可通过证明?AEB??FED，?ADF??GCF解决( (2013贵州贵阳，21，10分)如图，在正方形ABCD中，等边三角形AEF的顶点E,F分别在BC和CD上. (1)求证:CE=CF; (2)若等边三角形AEF的边长为2，求正方形ABCD的周长. A D F BC E 第21题图 解析:(1)可证Rt?ABE?Rt?ADF;(2)可得?EFC是等腰直角三角形，由等边三角形 长为2，可得EF=2，解直角三角形可得正方形ABCD的边长. AEF的边 解:(1)证明:?四边形ABCD正方形，??B=?D=90?,AB=AD. ??AEF是等边三角形，?AE=AF. ?Rt?ABE?Rt?ADF, ?BE=DF, ?BC=CD, ?CE=CF. 2(2)在Rt?EFC中，CE=CF=2×sin45?=. 2,62222设正方形ABCD的边长为x，则x+(x-)=2.解得，x=(舍负)，正方形ABCD2 2，662的周长为4×=2+2. 2 点评:直线型问题主要有两种形式，一种是证明，一种是计算，主要考查学生的逻辑推理能力以及空间观念.计算时一般考虑勾股定理、特殊角等的运用，列方程求解是常用方法. 23.4梯形 (2013广州市，5， 3分)如图2，在等腰梯形ABCD中，BC?AD，AD=5，DC=4，DE?AB交BC于点E，且EC =3，则梯形ABCD的周长是( ) plete, bolts should be tightened. 4.1.11.5 explosion dangerous environment of cable cannot directly connection, in nonequirements, may not be altered. Spiral bulbs should be tightened up and good contact, not loose, lamp housings should be comproof lamps lamp type, model and power, the design and specification of r-be less than 2mm. 4.1.11.4 the installation of explosion c gasket, metal gasket should be complete and should tighten the clamping larva inlet seals. Metal gasket thickness shall notelasti proof electrical equipment into the line port and wiring and cables, wires should be capable of reliably sealing, extra inlet-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-sfactory operation of explosionmeet the following requirements: repair and decoration works should end second grouting and finishing work should end, satiwithin explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall erials should be used to find the connection. 4.1.11.6 explosiveproof requirements should be to cope with different environmental conditions of different mat-proof electric junction box, explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-of explosionshall not is less than 40mm. Introduction burning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is less than pipe diameter, and-set fire wall, protection tube ends of tube mouth at, should will cable around with nonwall Shi, must take filling sand, and fill resistance fire blocking material or added proof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -tion Xia, must in corresponding of explosionnormal situa-6 AD CBE A. 26 B. 25 C. 21 D.20 【解析】由题意知，四边形ABED为平行四边形，可知BE=AD=5，从而得到BC的长， 【答案】梯形ABCD的周长为AB+BC+CD+AD=4+8+4+5=21.答案为C。 【点评】本题主要用到梯形常用的辅助线，把等腰梯形分为平行四边形和等腰三角形。关键是求出下底的长。 (2013山东省临沂市，11，3分)如图，在等腰梯形ABCD中，AD?BC,对角线AC、BD相交于点O，下列结论不一定正确的是( ) A.AC=BD B. OB=OC C. ?BCD=?BDC D. ?ABD=?ACD 【解析】?四边形ABCD是梯形,AD?BC,?AC=BD，?ABC=?DCB,?AOD?BOC,?OB=OC,?OBC=?OCB,?ABD=?DCA. 【答案】选C. 【点评】此题考查了等腰梯形的性质与相似三角形的判定与性质(解此题的关键是注意数形结合思想的应用与排除法的应用( (2013四川内江，16，5分)如图8，四边形ABCD是梯形，BD,AC且BD?AC，若AB,2，CD,4，则S, . 梯形ABCD A B C D 图8 【解析】如下图所示，过点B作BE?AC，与DC的延长线交于点E，BF?DE于F(接下 1111来，可证得?BDE是等腰直角三角形，BF,DE,(DC，CE),(DC，AB),(2，4),3，2222 11所以S,( AB，DC)?BF,(2，4)?3,9( 梯形ABCD22 -proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-s than pipe diameter, and shall not is less than 40mm. Introduction of explosionburning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is les-uld will cable around with none filling sand, and fill resistance fire blocking material or added set fire wall, protection tube ends of tube mouth at, showall Shi, must takproof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -normal situation Xia, must in corresponding of explosion-environment of cable cannot directly connection, in nonrous ghtened up and good contact, not loose, lamp housings should be complete, bolts should be tightened. 4.1.11.5 explosion dangelamps lamp type, model and power, the design and specification of requirements, may not be altered. Spiral bulbs should be ti proof-ten the clamping larva inlet seals. Metal gasket thickness shall not be less than 2mm. 4.1.11.4 the installation of explosionles, wires should be capable of reliably sealing, extra inlet elastic gasket, metal gasket should be complete and should tighproof electrical equipment into the line port and wiring and cab-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-works should end second grouting and finishing work should end, satisfactory operation of explosionmeet the following requirements: repair and decoration within explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall losivebe to cope with different environmental conditions of different materials should be used to find the connection. 4.1.11.6 exp proof requirements should-proof electric junction box, explosion7 A B O D E C F 【答案】9 【点评】在等腰梯形问题中，如果有对角线互相垂直条件，将其中一条对角线进行平移产生辅助线是常用解题思路(事实上，对角线互相垂直的等腰梯形的高等于其上、下底和的一半(解决此题，还可以证明?AOB和?COD是等腰直角三角形，在求得AC、BC长后，利用 1S,?ACD，?ACB,AC?BD解答( 梯形ABCD2 (2013四川省南充市，17，6分) 如图，等腰梯形ABCD中，AD?BC，点E是AD延长线上的一点，且CE=CD. 求证:?B=?E. 解析:先利用等腰三角形等边对等角推得?CDE=?E。根据AD?BC,可得?CDE=?DCB，等量代换得到?E=?DCB，再根据等腰梯形性质可知?B=?DCB，从而证得?B=?E。 答案:证明:?CE=CD， ??CDE=?E. ?AD?BC， ??CDE=?DCB. ??E=?DCB. ?AB=DC， ??B=?DCB. ??B=?E. 点评:本题主要考查等腰梯形的性质、等腰三角形的性质，及平行线性质。对于等腰梯形、等腰三角形内的角度问题，要充分利用底角相等的特点，再利用等量代换的方法即可探寻到所要求证角的相等关系。 plete, bolts should be tightened. 4.1.11.5 explosion dangerous environment of cable cannot directly connection, in nonequirements, may not be altered. Spiral bulbs should be tightened up and good contact, not loose, lamp housings should be comproof lamps lamp type, model and power, the design and specification of r-be less than 2mm. 4.1.11.4 the installation of explosion c gasket, metal gasket should be complete and should tighten the clamping larva inlet seals. Metal gasket thickness shall notelasti proof electrical equipment into the line port and wiring and cables, wires should be capable of reliably sealing, extra inlet-proof ventilation system should comply with the design requirements and. 4.1.11.3 explosion-sfactory operation of explosionmeet the following requirements: repair and decoration works should end second grouting and finishing work should end, satiwithin explosion and fire hazard atmospheres before the installation is put into operation, building installation work shall erials should be used to find the connection. 4.1.11.6 explosiveproof requirements should be to cope with different environmental conditions of different mat-proof electric junction box, explosion-proof motor cable lines looking for connections, can be used to find connections, between it and the explosion-of explosionshall not is less than 40mm. Introduction burning sex fiber jam closely, again filling sealed clay, sealed clay filling depth shall not is less than pipe diameter, and-set fire wall, protection tube ends of tube mouth at, should will cable around with nonwall Shi, must take filling sand, and fill resistance fire blocking material or added proof wiring box or points line box within connection or tap, cable line through different dangerous regional or territories -tion Xia, must in corresponding of explosionnormal situa-8