如图,在Rt△ABC中,∠A=90°,O是BC边上一点,以O为圆心的半圆与AB边相切于点D,与AC、BC边分别交于点E
如图,在Rt△ABC中,∠A=90°,O是BC边上一点,以O为圆心的半圆与AB边相切于点D,与AC、BC边分别交于点E、F、G,连接OD,已知BD=2,AE=3,tan∠...
如图,在Rt△ABC中,∠A=90°,O是BC边上一点,以O为圆心的半圆与AB边相切于点D,与AC、BC边分别交于点E、F、G,连接OD,已知BD=2,AE=3,tan∠BOD= . (1)求⊙O的半径OD;(2)求证:AE是⊙O的切线;(3)求图中两部分阴影面积的和.
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(1)3;(2)证明见解析;(3) ![](https://iknow-pic.cdn.bcebos.com/960a304e251f95ca9c53cf04ca177f3e660952d9?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . |
试题分析:(1)由AB为圆O的切线,利用切线的性质得到OD垂直于AB,在直角三角形BDO中,利用锐角三角函数定义,根据tan∠BOD及BD的值,求出OD的值即可; (2)连接OE,由AE=OD=3,且OD与AE平行,利用一组对边平行且相等的四边形为平行四边形,根据平行四边形的对边平行得到OE与AD平行,再由DA与AE垂直得到OE与AC垂直,即可得证; (3)阴影部分的面积由三角形BOD的面积+三角形ECO的面积-扇形DOF的面积-扇形EOG的面积,求出即可. 试题解析:(1)∵AB与圆O相切, ∴OD⊥AB, 在Rt△BDO中,BD=2,tan∠BOD= ![](https://iknow-pic.cdn.bcebos.com/35a85edf8db1cb137dcf61a1de54564e93584bd9?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴OD=3; (2)连接OE, ∵AE=OD=3,AE∥OD, ∴四边形AEOD为平行四边形, ∴AD∥EO, ∵DA⊥AE, ∴OE⊥AC, 又∵OE为圆的半径, ∴AE为圆O的切线; (3)∵OD∥AC, ∴ ![](https://iknow-pic.cdn.bcebos.com/48540923dd54564eaac3b98bb0de9c82d0584fd9?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/5ab5c9ea15ce36d33bbede8239f33a87e850b1ce?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) , ∴AC=7.5, ∴EC=AC-AE=7.5-3=4.5, ∴S 阴影 =S △BDO +S △OEC -S 扇形FOD -S 扇形EOG = ![](https://iknow-pic.cdn.bcebos.com/c8ea15ce36d3d539598752713987e950342ab0ce?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ×2×3+ ![](https://iknow-pic.cdn.bcebos.com/c8ea15ce36d3d539598752713987e950342ab0ce?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ×3×4.5- =3+ ![](https://iknow-pic.cdn.bcebos.com/5243fbf2b21193134af9a20266380cd790238dce?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/960a304e251f95ca9c53cf04ca177f3e660952d9?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) . 考点: 1.切线的判定与性质;2.扇形面积的计算. |
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