Question

一条船在海面上自西向东沿直线航行，在A处测得航标C在北偏东60°方向上，前进100米到达B处，又测得航标C在北偏东45°方向上．                                                                                  

（1）请根据以上描述，画出图形．                                                         

（2）已知以航标C为圆心，120米为半径的圆形区域内有浅滩，若这条船继续前进，是否有被浅滩阻碍的危险？为什么？                                                                       

                                                                     

Answer 1

【考点】解直角三角形的应用-方向角问题．                                           

【专题】计算题；压轴题．                                                                     

【分析】先根据题意画图，作CD⊥直线AB于点D，根据已知求得CD的长再与120比较，若大于120则无危险，反之有危险．                                                                                                   

【解答】解：（1）                                                                            

                                                                     

                                                                                                          

（2）答：这条船继续前进，没有被浅滩阻碍的危险．                                    

作CD⊥直线AB于点D，                                                                        

由已知可得∠CAD=30°，∠CBD=45°，                                                    

AB=100米．                                                                                      

设CD=x米．                                                                                     

在Rt△ACD中，tan∠CAD=，                                                             

∴AD=，                                                               

在Rt△CBD中，∵∠CBD=45°，                                                             

∴BD=CD=x，                                                                                    

∵AD﹣BD=AB，                                                                                

∴，                                                                             

解得，                                                                       

∴这条船继续前进没有被浅滩阻碍的危险．                                             

【点评】此题考查学生基本的作图能力，及构造直角三角形解题的能力．