(满分14分)如图,在△ABC中,AB=AC,以AB为直径的⊙O分别交BC、AC于D、E两点,过点D作DF⊥AC,垂足为点
(满分14分)如图,在△ABC中,AB=AC,以AB为直径的⊙O分别交BC、AC于D、E两点,过点D作DF⊥AC,垂足为点F.小题1:(1)求证:DF是⊙O的切线;小题2...
(满分14分)如图,在△ABC中,AB=AC,以AB为直径的⊙O分别交BC、AC于D、E两点,过点D作DF⊥AC,垂足为点F. 小题1:(1)求证:DF是⊙O的切线;小题2:(2)若弧AE=弧DE,DF=2,求弧AD的长.
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2014-10-21
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小题1:(1)证明:连结OD ∵AB=AC,∴∠C=∠B.………………………………………………2分 ∵OD=OB,∴∠B=∠1. ∴∠C=∠1. ………………………………………………………3分 ∴OD∥AC,∴∠2=∠FDO. …………………………… …………4分 ∵DF⊥AC,∴∠2=90°,∴∠FDO=90°, 即FD⊥OD且D点在⊙O 上---------------------------------5分 ∴FD是圆O的切线.……………………………………6分 小题2:(2)∵AB是⊙O的直径,∴∠ADB=90°.…………7分 ∵AC=AB,∴∠3=∠4………………………………8分 ∴ ![](https://iknow-pic.cdn.bcebos.com/9c16fdfaaf51f3dea702f7d697eef01f3b2979cb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∵ ![](https://iknow-pic.cdn.bcebos.com/ae51f3deb48f8c5481638b9d39292df5e1fe7fcb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴ ![](https://iknow-pic.cdn.bcebos.com/0df3d7ca7bcb0a46375fc58b6863f6246b60af07?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ……………9分 ∴∠B=2∠4,∴∠B=60°,∠5=120°, ∴△ABC是等边三角形, ∠C=60°.………………………11分 在Rt△CFD中,sinC= ![](https://iknow-pic.cdn.bcebos.com/d6ca7bcb0a46f21f784910e1f5246b600c33ae07?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,CD= ![](https://iknow-pic.cdn.bcebos.com/7acb0a46f21fbe09ac238da668600c338744ad07?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴DB= ![](https://iknow-pic.cdn.bcebos.com/8d5494eef01f3a2935789b7c9a25bc315d607ccb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,AB=BC= ![](https://iknow-pic.cdn.bcebos.com/bf096b63f6246b60c673fcc6e8f81a4c510fa207?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ∴AO= ![](https://iknow-pic.cdn.bcebos.com/8d5494eef01f3a2935789b7c9a25bc315d607ccb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) .……………………13分 ∴ ![](https://iknow-pic.cdn.bcebos.com/95eef01f3a292df5f973e2a7bf315c6035a873cb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) .………………………………………14分 |
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