如图,AB是⊙O的直径,弦CD⊥AB于点G,点F是CD上一点,且满足 = ,连接AF并延长交⊙O于点E,连接AD、DE
如图,AB是⊙O的直径,弦CD⊥AB于点G,点F是CD上一点,且满足=,连接AF并延长交⊙O于点E,连接AD、DE,若CF=2,AF=3.给出下列结论:①△ADF∽△AE...
如图,AB是⊙O的直径,弦CD⊥AB于点G,点F是CD上一点,且满足 = ,连接AF并延长交⊙O于点E,连接AD、DE,若CF=2,AF=3.给出下列结论:①△ADF∽△AED;②FG=2;③tan∠E= ;④S △ DEF = .其中正确的是 (写出所有正确结论的序号).
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王者38268
2014-09-22
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试题分析:①由AB是⊙O的直径,弦CD⊥AB,根据垂径定理可得: ![](https://iknow-pic.cdn.bcebos.com/377adab44aed2e73f15f209e8401a18b86d6faca?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/4bed2e738bd4b31cfb8a320984d6277f9f2ff8ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,DG=CG,继而证得△ADF∽△AED; ②由 ![](https://iknow-pic.cdn.bcebos.com/503d269759ee3d6dcbba217940166d224f4ade00?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/b21c8701a18b87d658f40dad040828381e30fdca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,CF=2,可求得DF的长,继而求得CG=DG=4,则可求得FG=2; ③由勾股定理可求得AG的长,即可求得tan∠ADF的值,继而求得tan∠E= ![](https://iknow-pic.cdn.bcebos.com/d6ca7bcb0a46f21f1048f8e1f5246b600c33ae00?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ; ④首先求得△ADF的面积,由相似三角形面积的比等于相似比,即可求得△ADE的面积,继而求得S △ DEF = ![](https://iknow-pic.cdn.bcebos.com/9922720e0cf3d7cad04099c4f11fbe096a63a9ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . ①∵AB是⊙O的直径,弦CD⊥AB, ∴ ![](https://iknow-pic.cdn.bcebos.com/377adab44aed2e73f15f209e8401a18b86d6faca?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/4bed2e738bd4b31cfb8a320984d6277f9f2ff8ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,DG=CG, ∴∠ADF=∠AED, ∵∠FAD=∠DAE(公共角), ∴△ADF∽△AED; 故①正确; ②∵ ![](https://iknow-pic.cdn.bcebos.com/503d269759ee3d6dcbba217940166d224f4ade00?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/b21c8701a18b87d658f40dad040828381e30fdca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,CF=2, ∴FD=6, ∴CD=DF+CF=8, ∴CG=DG=4, ∴FG=CG﹣CF=2; 故②正确; ③∵AF=3,FG=2, ∴AG= ![](https://iknow-pic.cdn.bcebos.com/9f2f070828381f306daf2b1baa014c086f06f0ca?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/7af40ad162d9f2d3c7a7825daaec8a136227ccca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴在Rt△AGD中,tan∠ADG= ![](https://iknow-pic.cdn.bcebos.com/91529822720e0cf37d8be8490946f21fbe09aa00?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/d6ca7bcb0a46f21f1048f8e1f5246b600c33ae00?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴tan∠E= ![](https://iknow-pic.cdn.bcebos.com/d6ca7bcb0a46f21f1048f8e1f5246b600c33ae00?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ; 故③错误; ④∵DF=DG+FG=6,AD= ![](https://iknow-pic.cdn.bcebos.com/9f2f070828381f306daf2b1baa014c086f06f0ca?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/29381f30e924b8992f8adf8a6d061d950b7bf6ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴S △ ADF = ![](https://iknow-pic.cdn.bcebos.com/6f061d950a7b0208f77f995361d9f2d3562cc8ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) DF?AG= ![](https://iknow-pic.cdn.bcebos.com/6f061d950a7b0208f77f995361d9f2d3562cc8ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ×6× ![](https://iknow-pic.cdn.bcebos.com/7af40ad162d9f2d3c7a7825daaec8a136227ccca?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/f7246b600c3387445fb989ce520fd9f9d72aa000?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∵△ADF∽△AED, ∴ ![](https://iknow-pic.cdn.bcebos.com/f3d3572c11dfa9ec1998f0a561d0f703908fc1ca?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/8644ebf81a4c510f6eb844a86359252dd42aa500?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/3c6d55fbb2fb4316cca9dcc823a4462308f7d3ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴S △ AED = ![](https://iknow-pic.cdn.bcebos.com/d52a2834349b033be4f45a6816ce36d3d539bd00?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴S △ DEF =S △ AED ﹣S △ ADF = ![](https://iknow-pic.cdn.bcebos.com/9922720e0cf3d7cad04099c4f11fbe096a63a9ca?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ; 故④正确. 故答案为:①②④. |
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