自主录题

自主录题一试题解答

试题解答

解：（1）如图①，连接OP，OM，过点O作OM'⊥AB，垂足为M'， 
<img alt="菁优网" src="http://img.jyeoo.net/quiz/images/svg/202306/613/376e90d7.png" style="vertical-align:middle" /> 
则 OP+PM≥OM． 
∵⊙O半径为4， 
∴PM≥OM-4≥OM'-4， 
∵OA=OB．∠AOB=120°， 
∴∠A=30°， 
∴OM'=AM'•tan30°=12tan30°=4<img latex="\sqrt{3}" src="data:image/png;base64,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" style="vertical-align: middle;" />， 
∴PM≥OM'-4=4<img latex="\sqrt{3}" src="data:image/png;base64,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" style="vertical-align: middle;" />-4， 
∴线段PM的最小值为4<img latex="\sqrt{3}" src="data:image/png;base64,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" style="vertical-align: middle;" />-4； 
（2）如图②，分别在BC，AE上作BB'=AA'=r=30（m）， 
<img alt="菁优网" src="http://img.jyeoo.net/quiz/images/svg/202306/613/12e98e61.png" style="vertical-align:middle" /> 
连接A'B'，B'O、OP、OE、B′E． 
∵OM⊥AB，BB'⊥AB，ON=BB'， 
∴四边形BB'ON是平行四边形． 
∴BN=B′O． 
∵B'O+OP+PE≥B'O+OE≥B'E， 
∴BN+PE≥B'E-r, 
∴当点O在B'E上时，BN+PE取得最小值． 
作⊙O'，使圆心O'在B'E上，半径r=30（m）， 
作O'M'⊥AB，垂足为M'，并与A'B'交于点H． 
∴O'H∥A'E， 
∴△B'O'H∽△B'EA'， 
∴<img latex="\frac{{O}^{\prime}H}{{E}^{}{A}^{\prime}}" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAgCAYAAAB6kdqOAAAAAXNSR0IArs4c6QAAAvVJREFUWEftWDty2zAQBUAdhC7TZDQEe0snidXkGrKvkUbySaT0BEeTJqV0EBLIPA5WA8K7pFRY4yJqbOH78Hbx8FZa3fhZLpfl6XS65MPRXhTFoe/7Ndd/4/LXYXpqQtxsq5Raaa3fQwjPSqky3byu61UIYeece0rXstYe4vcSfzG/aZrX2I6Dob3UWm+apjnSXBFQ3OigtV6nE+q6fg0h/CAA+I7FsFl+uHws9UdQZX6IATjHUGTmHELYtG27T8fkQLF43/cbLlzW2rNS6uic22TsBa31G3cIFhDR7Zxbc4CttcOCXdftkT/cSelQ+cZJ+4h5MWRVVb1orXccO5hEDKHfe39cLBZlGlJaOF3HGHO9DAi3UurFOceS8aHRWrvDhL7vn7gw0EZSPwGi/EEypyxHQIpjlc2hGHck3FQ4V1J/krh3548ECNeVvQHSzRO06Xxv/rCAItXb/LpjMJI9hPCe3zzhum/zsNLaU+x+CAspr1LqQteZ2kIIbzeAgVCSmK67rrsgF0lAo0huqD0/jKhDuD1QaEzw3l/mgGAcgBtjhjn4GGNKSEP8f9TOaZAojJz2PKpt8i17FIh0n/+A5lj/mgxVVfXNGPNzDv1n9nvvf7Vt+3dgCICUUt8/c8Mb1v5zBXTD4IcN+Zo5xB0fqhvVWmSH80HCQ7vNXSMpe25x5jz1YKaUUvsQwm88BViIzL7kaVJQkn+OD+1z7konQzb1OsM3zQEiuwKA+QsPG8y50klA+emwAb3ScJZcGHJ2YFdgiVMrQu7h5qqD4lsUxchkASBRDHBTORTLo6P3HrXXCNBU6TSXQ3CPKOJg0mEfLlIlkjITGQCDa85lAtDd9mOuIJRKawBLw5lWKeSppuaKDOW1GaoNlD3R/YknTBzntTzGTZXKqlwmJitXrrqkel9K6DTPaDMqLKUwzfohqbqg3JCMPkIFvcrtrlRSc4o7YiiqM+iFIEIEU9qHXytiYo9+6UgM/Kg/Cd8gqLHOZ8tzAvcP3XRmP/m81i4AAAAASUVORK5CYII=" style="vertical-align: middle;" />＝<img latex="\frac{{B}^{\prime}H}{{B}^{\prime}{A}^{\prime}}" src="data:image/png;base64,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" style="vertical-align: middle;" />， 
∵⊙O'在矩形AFDE区域内（含边界）， 
∴当⊙O'与FD相切时，B′H最短，即B′H=10000-6000+30=4030（m）． 
此时，O′H也最短． 
∵M'N'=O'H， 
∴M'N'也最短． 
∴O'H=<img latex="\frac{E{A}^{\prime}\cdot{B}^{\prime}H}{{B}^{\prime}{A}^{\prime}}" src="data:image/png;base64,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" style="vertical-align: middle;" /> ＝<img latex="\frac{(10000−30)\times4030}{10000}" src="data:image/png;base64,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" style="vertical-align: middle;" />=4017.91（m）， 
∴O'M'=O'H+30=4047.91（m）， 
∴此时环道⊙O的圆心O到AB的距离OM的长为4047.91m.

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