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www6629llcom:【学海导航】高考数学第1轮总复习 全国统编教材 9.2空间直线(第2课时)课件 理_图文

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第九章 直线、平面、简单几何体 第 讲 (第二课时) 题型4 求异面直线所成的角 1. 设P为正三角形ABC所在 平面外一点PA=PB=PC, ∠APB=∠BPC=∠CPA=90°, M、N分别是AB和PC的中点, 求异面直线PM与BN所成的角. 解:连结MC,取其中点D,连结ND, 1 则ND / / PM 所以∠BND为所求的角. 2 连结BD,设正三角形ABC的边长为a. 由已知,△APB,△BPC, △APC都是等腰直角三角 a 形,所以PM= 2 易知 BN ? 2 2 PB ? PN ? ( a) 2 2 2 10 ? a 4 a DN= . 4 又CM= 3 3 a a. ,所以 DM = 2 4 2 2 在Rt△BMD中, a 2 3 2 7 BD ? BM ? DM ? ( ) ? ( a) ? a 2 4 4 BN 2 ? DN 2 ? BD 2 10 cos ? BND ? ? 在△BND中, 2 BN ? DN 5 10 故异面直线PM与BN所成的角为arccos . 5 点评:求异面直线所成的角的关键 是作辅助线来平移直线,化为同一平面 内两直线所成的角 . 一般根据中点可作 中位线平移直线,或由平行四边形的性 质平移直线,然后利用解三角形的有关 知识求得夹角. 如图,在正方体ABCD-A1 B1 C1 D1中, E、F分别是BB1、CD的中点. (1)证明:AD⊥D1F; (2)求AE与D1F所成的角. 解:(1)证明:因为ABCDA1B1C1D1是正方体, 所以AD⊥平面DCC1D1. 又DF1 ? 平面DCC1D1,所以AD⊥D1F. (2)取AB的中点G,连结A1G,FG, 因为F是CD的中点, 所以 GF / / A1 D1 又 A1 D1 / / AD 所以GF / / AD , 故四边形GFD1A1是平行四边形, 所以A1G∥D1F. 设A1G与AE相交于H, 则∠A1HA是AE与D1F所成的角. 因为E是BB1的中点, 所以 Rt△A1AG≌Rt△ABE, 所以∠GA1A=∠EAB, 从而∠A1HA=90°, 即直线AE与D1F所成的角为90°. 题型4 异面直线夹角条件的转化 2. 已知正三棱柱ABC-A1B1C1的底边 长为8,异面直线AB1和BC1所成的角为 a r c cos 1 ,求这个正三棱柱的侧棱长. 25 解:连结B1C交BC1于D点, 则D为B1C的中点. 取AC的中点E,连结DE, 1 则 DE / / AB1 2 所以∠BDE为异面直线AB1 和BC1所成的角或其补角. 设正三棱柱的侧棱长为x. 因为正三棱柱的底边长为8, 则BE=8sin60°= 4 3 , AB1 ? BC1 ? x 2 ? 64 , 1 所以 BD ? DE ? x 2 ? 64 2 在△BDE中,BE2=BD2+DE22BD· DEcos∠BDE. 点评:已知角度求边长问题,一般是结 合方程思想来解决,即先设边长为参数,然 后根据题中条件转化为参数方程(组),再解 方程(组),即可求得边长的值.注意方程的解 与边长的值的实际意义. 1 若cos∠BDE= 25 , 1 2 1 2 则 48 ? 2 (x ? 64 ) ? 50 (x ? 64 ) 解得`x=6. 1 23 ? 若cos∠BDE= ,则x= 4 . 25 13 23 故这个正三棱柱的侧棱长为6或 4 . 13 如图,正方形ABCD的边长为 3,E、F分别是AB、CD上的点,且BE=1. 将四边形AEFD沿EF折起到A′EFD′的位 置,使异面直线EB和D′F所成 的角为60°.求证:点A′在 平面ABCD内的射影O恰好 在BC边上. 证明:因为A′E∥D′F, 所以∠A′EB为异面直线EB和D′F 所成的角,知∠A′EB=60°. 因为BE=1,A′E=AE=2,连结 A′B, 则在△A′EB中, A′B2=A′E2+BE2-2 A ′ E· BEcos60°=3, 所以A′B2+BE2=A′E2, 所以A′B⊥EB. 又EB⊥BC,所以EB⊥平面A′BC, 所以平面A′BC⊥平面ABCD. 据两平面垂直的性质,知点A′ 在平面ABCD内的射影O在BC边上. 1. 求异面直线所成的角大致可分四个 步骤进行:找出或作出异面直线所成的角 ( 或其补角 )→ 构造三角形 → 解三角形,即 求角的某个三角函数值→小结. 2. 作异面直线所成的角要用连线的办 法,先定位再定性,一般不要直接作平行 线 . 添加的辅助线要尽可能位于背景图形 “内部”,并在解题过程中加以说明. 3. 用余弦定理解三角形,若出现所 求角的余弦值为负值,则异面直线所成 的角要取其补角转化为锐角. 4. 若已知异面直线所成的角的大小, 求解其他问题,则先应找出或作出异面 直线所成的角,再通过解三角形将其转 化为边的关系.

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