如图,在△ABC中,AB=AC,以AB为直径作⊙O,交BC于点D,连接OD,过点D作⊙O的切线,交AB延长线于点E,交A
如图,在△ABC中,AB=AC,以AB为直径作⊙O,交BC于点D,连接OD,过点D作⊙O的切线,交AB延长线于点E,交AC于点F.(1)求证:OD∥AC;(2)当AB=1...
如图,在△ABC中,AB=AC,以AB为直径作⊙O,交BC于点D,连接OD,过点D作⊙O的切线,交AB延长线于点E,交AC于点F.(1)求证:OD∥AC;(2)当AB=10, 时,求AF及BE的长.
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神独谮锨善拘2
推荐于2017-10-09
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(1)证明见解析;(2) ![](https://iknow-pic.cdn.bcebos.com/c8177f3e6709c93d399349459c3df8dcd100541a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . |
试题分析:(1)若要证明OD∥AC,则可转化为证明∠C=∠ODB即可. (2)连接AD,首先利用已知条件可求出BD的长,再证明△ADC∽△AFD,利用相似三角形的性质:对应边的比值相等即可求出AF及BE的长. 试题解析:(1)∵AB=AC,∴∠ABC=∠C. ∵OB=OD,∴∠OBD=∠ODB.∴∠C=∠ODB. ∴OD∥AC. (2)如图,连接AD, ∵AB为直径,∴AB⊥BD.∴∠ADC=90°. ∵AB=10, ![](https://iknow-pic.cdn.bcebos.com/574e9258d109b3deaddea3dacfbf6c81800a4c1a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,∴BD=AB?cos∠ABC= ![](https://iknow-pic.cdn.bcebos.com/9358d109b3de9c82e304be3d6f81800a19d8431a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) .∴AD= ![](https://iknow-pic.cdn.bcebos.com/838ba61ea8d3fd1f7f68e588334e251f95ca5f1a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ∵DF是圆的切线,∴OD⊥DF.∴∠ODF=90°. ∵AC∥OD,∴∠AFD=90°. ∵∠ADC=∠AFD,∠DAF=∠CAD,∴△ADC∽△AFD, ∴ ![](https://iknow-pic.cdn.bcebos.com/a71ea8d3fd1f4134a85642cc261f95cad1c85e1a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,即 ![](https://iknow-pic.cdn.bcebos.com/4034970a304e251fd796a34aa486c9177f3e531a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,解得AF=8. ∵OD∥AF,∴ ![](https://iknow-pic.cdn.bcebos.com/960a304e251f95ca9294d504ca177f3e6709521a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,即 ![](https://iknow-pic.cdn.bcebos.com/a6efce1b9d16fdfab5e1815cb78f8c5495ee7bf6?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ∴BE= ![](https://iknow-pic.cdn.bcebos.com/c8177f3e6709c93d399349459c3df8dcd100541a?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . |
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