中考数学考前试练2

1 中考数学考前试练 2 操作型数学试题专练 【辽宁省沈阳市】24. 如图 1，在△ABC 中，点 P 为 BC 边中点，直线 a 绕顶点 A 旋转， 若 B、P 在直线 a 的异侧， BM直线 a 于点 M，CN直线 a 于点 N，连接 PM、PN； （1） 延长 MP 交 CN 于点 E（如图 2）。 求证：△BPM△CPE； 求证：PM = PN； （2） 若直线 a 绕点 A 旋转到图 3 的位置时，点 B、P 在直线 a 的同侧，其它条件不变。 此时 PM=PN 还成立吗？若成立，请给予证明；若不成立，请说明理由； （3） 若直线 a 绕点 A 旋转到与 BC 边平行的位置时，其它条件不变。请直接判断四边 形 MBCN 的形状及此时 PM=PN 还成立吗？不必说明理由。 【河北省】24．（本小题满分 10 分） 在图 15-1 至图 15-3 中，直线 MN 与线段 AB 相交于点 O，∠1 = ∠2 = 45°． （1）如图 15-1，若 AO = OB，请写出 AO 与 BD 的数量关系和位置关系； （2）将图 15-1 中的 MN 绕点 O 顺时针旋转得到图 15-2，其中 AO = OB． 求证：AC = BD，AC ⊥ BD； （3）将图 15-2 中的 OB 拉长为 AO 的 k 倍得到图 15-3，求 AC BD 的值． aA B CP M N A B C M N a P A B CP N M a 圖 1 圖 2 圖 3 图 15-1 A D B M N 1 2 O 图 15-2 A D O B C 2 1 M N 图 15-3 A D O BC 2 1 M N 2 【湖南省常德市】 26.如图 10，若四边形 ABCD、四边形 CFED 都是正方形，显然图中有 AG=CE，AG⊥CE. （1）当正方形 GFED 绕 D 旋转到如图 11 的位置时，AG=CE 是否成立？若成立，请给出证 明；若不成立，请说明理由. （2）当正方形 GFED 绕 D 旋转到如图 12 的位置时，延长 CE 交 AG 于 H，交 AD 于 M. ①求证：AG⊥CH;②当 AD=4，DG= 2 时，求 CH 的长。 二十八、【无锡市】 26．【无锡市】（本题满分 10 分） （1）如图 1，在正方形 ABCD 中，M 是 BC 边（不含端点 B、C）上任意一点,P 是 BC 延长线上一点，N 是∠DCP 的平分线上一点．若∠AMN=90°，求证：AM=MN． 下面给出一种证明的思路，你可以按这一思路证明，也可以选择另外的方法证明． 证明：在边 AB 上截取 AE=MC，连 ME．正方形 ABCD 中，∠B=∠BCD=90°， AB=BC．∴∠NMC=180°—∠AMN—∠AMB=180°—∠B—∠AMB=∠MAB =∠MAE．（下面请你完成余下的证明过程） （2）若将（1）中的“正方形 ABCD”改为“正三角形 ABC”（如图 2）,N 是∠ACP 的平分线上一点，则 当∠AMN=60°时，结论 AM=MN 是否还成立？请 说明理由． （3）若将（1）中的“正方形 ABCD”改为“正 n 边形 ABCD…X”，请你作出猜想：当∠AMN = °时，结论 AM=MN 仍然成立．（直接 A B C D EF 图 110 G A D 图 11 F E B C G A D B C E F H M 图 12 图 1 图 2 3 A B C NM P A M N P1 CP2 B A C M N P1 P2 P2009…… …… B 第 23 题图 2第 23 题图 1 第 23 题图 3 写出答案，不需要证明） 【浙江省绍兴市】23. （1） 如图 1,在正方形 ABCD 中,点 E,F 分别在边 BC,CD 上,AE,BF 交于点 O,∠AOF＝90°.求证：BE＝CF. （2） 如图 2,在正方形 ABCD 中,点 E,H,F,G 分别在边 AB, BC,CD,DA 上,EF,GH 交于点 O,∠FOH＝90°, EF＝4.求 GH 的长. （3）已知点E,H,F,G分别在矩形ABCD的边AB,BC,CD,DA上，EF,GH 交于点 O,∠FOH＝90°,EF＝4. 直接写出下列两题的答案： ①如图 3,矩形 ABCD 由 2 个全等的正方形组成,求 GH 的长； ②如图 4,矩形 ABCD 由 n 个全等的正方形组成,求 GH 的长（用 n 的代数式表示）. BA CD F E OG H 第 23 题图 3 BA CD F E OG H 第 23 题图 4 23．【济南市】已知：△ABC 是任意三角形． ⑴如图 1 所示，点 M、P、N 分别是边 AB、BC、CA 的中点．求证：∠MPN=∠A． ⑵如图 2 所示，点 M、N 分别在边 AB、AC 上，且 1 3 AM AB  ， 1 3 AN AC  ，点 P1、P2 是边 BC 的三等分点，你认为∠MP1N+∠MP2N=∠A 是否正确？请说明你的理由． ⑶【济南市 1 2010 AN AC  ，点 P1、P2、……、P2009 是边 BC 的 2010 等分点，则∠MP1N+ ∠MP2N+……+∠MP2009N=____________． （请直接将该小问的答案写在横线上．） BA C C D F F EO O BA C C D F F H O O E G 4 内部资料， 请勿外传！ 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