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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">问题提出</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（1）如图①，△ABC是等边三角形，AB=12，若点O是△ABC的内心，则OA的长为<u> </u></span></span></span></span></span><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">；</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">问题探究</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（2）如图②，在矩形ABCD中，AB=12，AD=18，如果点P是AD边上一点，且AP=3，那么BC边上是否存在一点Q，使得线段PQ将矩形ABCD的面积平分？若存在，求出PQ的长；若不存在，请说明理由．</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">问题解决</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（3）某城市街角有一草坪，草坪是由△ABM草地和弦AB与其所对的劣弧围成的草地组成，如图③所示．管理员王师傅在M处的水管上安装了一喷灌龙头，以后，他想只用喷灌龙头来给这块草坪浇水，并且在用喷灌龙头浇水时，既要能确保草坪的每个角落都能浇上水，又能节约用水，于是，他让喷灌龙头的转角正好等于∠AMB（即每次喷灌时喷灌龙头由MA转到MB，然后再转回，这样往复喷灌．）同时，再合理设计好喷灌龙头喷水的射程就可以了．</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">如图③，已测出AB=24m，MB=10m，△AMB的面积为96m<sup>2</sup>；过弦AB的中点D作DE⊥AB交</span></span></span></span></span><img height="20" src="/uploads/meth/20210206/98e27ec12ec3388d004ce812913417e3.png" width="18" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">于点E，又测得DE=8m．</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">请你根据以上信息，帮助王师傅计算喷灌龙头的射程至少多少米时，才能实现他的想法？为什么？（结果保留根号或精确到0.01米）</span></span></span></span></span><br /> <img height="175" src="/uploads/meth/20210206/718f85cf97d7f482c8cc12b4c4818f85.png" width="442" />
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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">解：（1）如图1，过O作OD⊥AC于D，则AD=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/9bface64b1f8b00fb3114ef85bb7ba04.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">AC=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/3247cce0ac732cfb16cdce0c59a5c094.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">×12=6，</span></span></span></span></span><br /> <img height="113" src="/uploads/meth/20210206/c7720f77714eee294d0e0403a31fbc4e.png" width="113" /><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵O是内心，△ABC是等边三角形，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴∠OAD=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/c4c4d47b79b2c68b630e280d1f8666ec.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∠BAC=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/0d0cc24453b4112107172bd533a25b39.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">×60°=30°，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">在Rt△AOD中，cos∠OAD=cos30°=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/1dab86ba9fa0896d3147c2dd34c07cb1.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴OA=6÷</span></span></span></span></span><img height="37" src="/uploads/meth/20210206/2d83ff7b31ad57c4d8850f460031c19b.png" width="23" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">=4</span></span></span></span></span><img height="18" src="/uploads/meth/20210206/77b3575d18e24ec905b7e920c294b04c.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">故答案为：4</span></span></span></span></span><img height="18" src="/uploads/meth/20210206/1034bd3408f0633a5126c0559aa23062.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">；</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（2）存在，如图2，连接AC、BD交于点O，连接PO并延长交BC于Q，则线段PQ将矩形ABCD的面积平分，</span></span></span></span></span><br /> <img height="131" src="/uploads/meth/20210206/4feb36ffcdc344f674cc4e7af3876ea2.png" width="129" /><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵点O为矩形ABCD的对称中心，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴CQ=AP=3，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">过P作PM⊥BC于点M，则PM=AB=12，MQ=18﹣3﹣3=12，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">由勾股定理得：PQ=</span></span></span></span></span><img height="24" src="/uploads/meth/20210206/98f91287ee079dfb813243744b8b81ea.png" width="75" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">=</span></span></span></span></span><img height="24" src="/uploads/meth/20210206/37e02d5dbb29f7962697854bbb7bb211.png" width="75" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">=12</span></span></span></span></span><img height="18" src="/uploads/meth/20210206/8f2a734c3270d204b0bd7dacb0a931a3.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">；</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（3）如图3，作射线ED交AM于点C</span></span></span></span></span><br /> <img height="163" src="/uploads/meth/20210206/9f7e73b6862b90c734739899a507c586.png" width="174" /><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵AD=DB，ED⊥AB，</span></span></span></span></span><img height="20" src="/uploads/meth/20210206/4aaa39f8cd079080f906b63a430a5e92.png" width="18" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">是劣弧，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴</span></span></span></span></span><img height="20" src="/uploads/meth/20210206/3d61c0c953ac19daa66675ce596fc7db.png" width="18" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">所在圆的圆心在射线DC上，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">假设圆心为O，半径为r，连接OA，则OA=r，OD=r﹣8，AD=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/f58ad80bde2156dc74568f30ac819864.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">AB=12，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">在Rt△AOD中，r<sup>2</sup>=12<sup>2</sup>+（r﹣8）<sup>2</sup>，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">解得：r=13，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴OD=5，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">过点M作MN⊥AB，垂足为N，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵S<sub>△ABM</sub>=96，AB=24，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/4e2f35df2697aa179de7a4380f56f3d2.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">AB•MN=96，</span></span></span></span></span><br /> <img height="35" src="/uploads/meth/20210206/0fae31a2f925f28b4f19ba2ce02ad45f.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">×24×MN=96，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴MN=8，NB=6，AN=18，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵CD∥MN，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴△ADC∽△ANM，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/ff7791c4bb0d128e4968a978ec88dfaf.png" width="52" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/12592664df86fa949332132186118e21.png" width="52" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴DC=</span></span></span></span></span><img height="35" src="/uploads/meth/20210206/a225432ff203b016118a524618509fb2.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴OD＜CD，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴点O在△AMB内部，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴连接MO并延长交</span></span></span></span></span><img height="20" src="/uploads/meth/20210206/80622871678d0c6d049ca1373f73474a.png" width="18" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">于点F，则MF为草坪上的点到M点的最大距离，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵在</span></span></span></span></span><img height="20" src="/uploads/meth/20210206/cc10df27c501e578a7c849ea35b9cd53.png" width="18" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">上任取一点异于点F的点G，连接GO，GM，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴MF=OM+OF=OM+OG＞MG，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">即MF＞MG，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">过O作OH⊥MN，垂足为H，则OH=DN=6，MH=3，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴OM=</span></span></span></span></span><img height="24" src="/uploads/meth/20210206/07c6bccececddd08fbef483d821f6da0.png" width="75" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">=</span></span></span></span></span><img height="24" src="/uploads/meth/20210206/31202528cdaf99b11b8cec66da85defa.png" width="57" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">=3</span></span></span></span></span><img height="18" src="/uploads/meth/20210206/c01857aa38961b1151e3e5c00edd3e1f.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴MF=OM+r=3</span></span></span></span></span><img height="18" src="/uploads/meth/20210206/346bcd2d2072bc256e424636de51bc35.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">+13≈19.71（米），</span></span></span></span></span><br /> <span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">答：喷灌龙头的射程至少为19.71米．</span></span></span></span></span>
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本题考查了"等边三角形的性质 勾股定理 矩形的性质 圆的综合题 特殊角的三角函数值 "，属于"压轴题"，熟悉知识点是解题的关键
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