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2017-2018学年人教A版高中数学选修4-1课件:第二讲 四 弦切角的性质_图文

四 弦切角的性质 弦切角定理 (1)文字语言叙述: 弦切角等于它 所夹的弧 所对的圆周角. (2)图形语言叙述: 如图,AB与⊙O切于A点,则∠BAC= ∠D . [说明] 弦切角的度数等于它所夹弧度数的一半,圆 周角的度数等于它所对的弧的度数的一半,圆心角的度数 等于它所对弧的度数. 弦切角定理 [例1] ? ,过C点的圆的切线与 AC = BD 如图,已知圆上的 ? BA的延长线交于E点. 证明:(1)∠ACE=∠BCD; (2)BC2=BE· CD. [思路点拨] 利用弦切角定理. [证明] ? , AC = BD (1)因为 ? 所以∠BCD=∠ABC. 又因为EC与圆相切于点C, 所以∠ACE=∠ABC. 所以∠ACE=∠BCD. (2)因为∠ECB=∠CDB,∠EBC=∠BCD, 所以△BDC∽△ECB. BC CD 故BE= BC, 即BC2=BE· CD. 利用弦切角定理进行计算、证明时,要特别注意弦切 角所夹弧所对的圆周角,有时与圆的直径所对的圆周角结 合运用,同时要注意根据题目的需要添加辅助线构造所需 要的弦切角. ? 上的一点,若 1.如图,CD是⊙O的切线,T为切点,A是 TB ∠TAB=100°,则∠BTD的度数为 ( ) A.20° B.40° C.60 ° D.80° 解析:如图,作四边形 ABET,因为四边形 ABET 是圆内 接四边形, 所以∠E=180°-∠TAB=80°. 又 CD 是⊙O 的切线,T 为切点, 所以∠BTD=∠E=80°. 答案:D 2.如图,AB是⊙O的弦,CD是经过⊙O上的点M 的切线,求证: (1)如果AB∥CD,那么AM=MB; (2)如果AM=BM,那么AB∥CD. 证明:(1)∵CD切⊙O于M点, ∴∠CMA=∠B. ∵AB∥CD, ∴∠CMA=∠A. ∴∠A=∠B. ∴AM=MB. (2)∵AM=BM, ∴∠A=∠B. ∵CD切⊙O于M点,∠CMA=∠B, ∴∠CMA=∠A. ∴AB∥CD. 3.如图,已知AB是⊙O的直径,直线CD与⊙O相切 于点C,AC平分∠DAB. (1)求证:AD⊥CD; (2)若AD=2,AC= 5,求AB的长. 解:(1)证明:如图,连接BC. ∵直线CD与⊙O相切于点C, ∴∠DCA=∠B. ∵AC平分∠DAB, ∴∠DAC=∠CAB. ∴∠ADC=∠ACB. ∵AB为⊙O的直径, ∴∠ACB=90°. ∴∠ADC=90°,即AD⊥CD. (2)∵∠DCA=∠B,∠DAC=∠CAB, ∴△ADC∽△ACB. AD AC ∴ AC=AB, ∴AC2=AD· AB. ∵AD=2,AC= 5, 5 ∴AB=2. 运用弦切角定理证明比例式或乘积式 [例 2] 如图,PA,PB 是⊙O 的切线,点 AB 上,CD⊥AB,CE⊥PA,CF⊥PB, C 在? 垂足分别为 D, E,F. 求证:CD2=CE· CF. [思路点拨] 连接CA,CB, Rt△CAE∽Rt△CBD ∠CAP=∠CBA, ―→ Rt△CBF∽Rt△CAD ∠CBP=∠CAB CE CD → CD= CF → 结论 [证明] 连接CA,CB. ∵PA,PB是⊙O的切线, ∴∠CAP=∠CBA,∠CBP=∠CAB. 又CD⊥AB,CE⊥PA,CF⊥PB, ∴Rt△CAE∽Rt△CBD, Rt△CBF∽Rt△CAD, CA CE CB CF ∴CB=CD,CA=CD. CE CD ∴CD= CF, 即CD2=CE· CF. 证明乘积式成立,往往与相似三角形有关,若存在 切线,常要寻找弦切角,确定三角形相似的条件,有时 需要添加辅助线创造条件. 4.如图,已知MN是⊙O的切线,A为切点,MN 平行于弦CD,弦AB交CD于点E. 求证:AC2=AE· AB. 证明:连接BC. MN∥CD?∠MAC=∠ACD ? ? ? MN切⊙O于A?∠MAC=∠B? ? ?∠ACD=∠B ? ? ??△ACE∽△ABC ∠CAE=∠CAB? ? AC AE ?AB=AC?AC2=AB· AE. 5.如图,AD是△ABC的角平分线,经过点 A,D的⊙O和BC切于点D,且AB,AC 与⊙O相交于点E,F,连接DF,EF. 求证:(1)EF∥BC; (2)DF2=AF· BE. 证明:(1)∵⊙O切BC于点D, ∴∠CAD=∠CDF. ∵AD是△ABC的角平分线, ∴∠BAD=∠CAD. 又∵∠BAD=∠EFD, ∴∠EFD=∠CDF. ∴EF∥BC. (2)连接DE. ∵⊙O切BC于D, ∴∠BAD=∠BDE. 由(1)可得∠BDE=∠FAD, 又∵⊙O内接四边形AEDF, ∴∠BED=∠DFA. ∴△BED∽△DFA. DE BE ∴ AF=DF. 又∵∠BAD=∠CAD, ∴DE=DF. ∴DF2=AF· BE. 课时跟踪检测见课时跟踪检测(九)

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