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wwwhg7651com:高中数学人教A版选修4-1课件:第二讲 一 圆周角定理_图文

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一 圆周角定理 1.圆周角定理 圆上一条弧所对的圆周角等于它所对的圆心角的 一半 . 应当注意的是,圆周角与圆心角一定是对着同一条弧,它 们才有上面定理中所说的数量关系. 2.圆心角定理 (1)圆心角的度数 等于 它所对的弧的度数,它与圆的半径 无关,也就是说在大小不等的两个圆中,相同度数的圆心角, 它们所对的弧的度数 相等 ;反过来,弧的度数相等,它们所对 的圆心角的度数 也相等 . AB )的度数相等,不能 (2)圆心角(∠AOB)与它所对的弧( ? AB ,正确写法是∠AOB的度数= ? AB 的度数. 写成∠AOB= ? 3.圆周角定理的推论 (1)推论1:同弧或等弧所对的圆周角 相等 ;同圆或等圆 中,相等的圆周角所对的弧也相等. [说明] (1)若将“同弧或等弧”改为“同弦或等弦”,则结论不 成立. (2)相等的弧与相同度数的弧含义是不同的.只有弧的度 数和弧的长度都相等的两条弧才是等弧,即等弧一定有相同 的度数,而有相同度数的弧不一定是等弧. (3)“相等的圆周角所对的弧也相等”的前提条件是“在 同圆或等圆中”,应用推论时要时刻记住这一点. (2)推论2:半圆(或直径)所对的圆周角是直角;90°的圆 周角所对的弦是直径. 与圆周角定理相关的证明 [例1] 已知:如图,△ABC内接于⊙O,D, E在BC边上,且BD=CE,∠1=∠2. 求证:AB=AC. [思路点拨] 证明此题可先添加辅助线构造等 弦、等弧的条件,再由圆周角定理及其推论证明. [证明] 如图,延长AD,AE分别交⊙O于 F,G两点,连接BF,CG, ? , ? = CG ∵∠1=∠2,∴ BF ? , ? = CF ∴BF=CG, BG ∴∠FBD=∠GCE. 又∵BD=CE, ∴△BFD≌△CGE, ∴∠F=∠G, AB = ? AC ,∴AB=AC. ∴? 利用圆周角定理证明等量关系时,主要是分析圆周 角、圆心角、弧、弦之间的等量关系,有时需添加辅助线 构造等弧、等角、等弦的条件. 1.已知AD是△ABC的高,AE是△ABC的外接圆的直径. 求证:∠BAE=∠DAC. 证明:连接BE, 因为AE为直径, 所以∠ABE=90°. 因为AD是△ABC的高, 所以∠ADC=90°. 所以∠ADC=∠ABE. 因为∠E=∠C, ∠BAE=90°-∠E, ∠DAC=90°-∠C. 所以∠BAE=∠DAC. 2.已知⊙O中,AB=AC,D是BC延长线上一点,AD交⊙O于 点E. 求证:AB2=AD· AE. AB = ? AC . 证明:如图,∵AB=AC,∴ ? ∴∠ABD=∠AEB. 在△ABE与△ADB中, ∠BAE=∠DAB, ∠AEB=∠ABD, ∴△ABE∽△ADB. AB AE ∴AD=AB,即AB2=AD· AE. 利用圆周角进行计算 [例2] 如图,已知BC为半⊙O的直径, AD⊥BC,垂足为D,BF交AD于点E,且AE= BE. AB = ? AF ; (1)求证: ? 3 (2)如果sin ∠FBC=5,AB=4 5,求AD的长. [思路点拨] ABC. BC为半⊙O的直径,连接AC,构造Rt△ [解] (1)证明:如图,连接AC. ∵BC是⊙O的直径, ∴∠BAC=90°, 又AD⊥BC,垂足为D, ∴∠BAD=∠ACB. 在△AEB中,AE=BE, ∴∠ABE=∠BAE. AB = ? AF . ∴∠ABF=∠ACB,即 ? (2)设DE=3x, 3 ∵AD⊥BC,sin∠FBC=5, ∴BE=5x,BD=4x. ∵AE=BE,∴AE=5x,AD=8x. 在Rt△ADB中,∠ADB=90°,AB=4 5, ∴(8x)2+(4x)2=(4 5)2, 解得x=1,∴AD=8. 与圆周角定理有关的线段的计算、角的计算,不仅可 以通过计算弧、圆心角、圆周角的度数来求相关的角、线 段,有时还可以通过三角形相似、解三角形等来计算. 3.如图,直径为10的⊙A经过点C(0,5)和点 O(0,0),B是y轴右侧⊙A弧上一点,则 cos∠OBC的值为 1 A.2 3 B. 2 ( ) 3 4 C.5 D.5 解析:法一:设⊙A与x轴另一个交点为D, 连接CD,如图所示. 因为∠COD=90°, 所以CD为⊙A的直径. 又因为∠CBO 与∠CDO 为圆弧 CO 所对 的圆周角, 所以∠CBO=∠CDO. 又因为 C(0,5), 所以 OC=5. 在 Rt△CDO 中,CD=10,CO=5, 根据勾股定理得 OD= CD2-OC2=5 3. OD 5 3 3 所以 cos∠OBC=cos∠CDO= CD= 10 = 2 ,故选 B. 法二:连接 AO,AC,因为 OC=5,AC= AO=5, 所以△ACO 为等边三角形, ∠CAO=60°, 1 ∠CBO=2∠CAO=30°, 3 所以 cos∠CBO=cos 30°= 2 . 答案:B AB = ? AC , 4.已知,如图,△ABC内接于⊙O, ? ? 上任意一点,AD与BC交于点E,AD 点D是 BC =6 cm,BD=5 cm,CD=3 cm,求DE的长. AB = ? AC , 解:∵ ? ∴∠ADB=∠CDE. ? = BD ? ,∴∠BAD=∠ECD. 又∵ BD ∴△ABD∽△CED. AD BD ∴CD=ED, 6 5 即3=ED. ∴ED=2.5 (cm). 5.如图,△ABC的角平分线AD的延长线交它 的外接圆于点E. (1)证明:△ABE∽△ADC; 1 (2)若△ABC的面积S=2AD· AE, 求∠BAC的大小. 解:(1)证明:由已知条件可得∠BAE=∠CAD. 因为∠AEB与∠ACB是同弧上的圆周角, 所以∠AEB=∠ACD. 故△ABE∽△ADC. (2)因为△ABE∽△ADC, AB AD 所以AE= AC,即AB· AC=AD· AE.

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