1.
(2024九上·北京市月考)
在平面直角坐标系
中,已知矩形OABC,其中点
, 给出如下定义:若点P关于直线
的对称点
在矩形OABC的内部或边上,则称点P为矩形OABC关于直线l的“关联点”.
![](//tikupic.21cnjy.com/ct20241o/a0/52/a052d50df4c9b59bf8d1bdcfd89d0144.png)
![](//tikupic.21cnjy.com/ct20241o/2a/4c/2a4cdba831b5e54b833876bac715be96.png)
例如,图1中的点D,点E都是矩形OABC关于直线
的“关联点”.
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(1)
如图2,在点
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E6%3C%2Fmn%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
中,是矩形OABC关于直线
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3El%3C%2Fmi%3E%3Cmi+mathvariant%3D%22normal%22%3E%EF%BC%9A%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的“关联点”的为_____________;
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(2)
如图3,点
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi+mathvariant%3D%22normal%22%3E++%EF%BC%8C+%3C%2Fmi%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
是矩形OABC关于直线
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3El%3C%2Fmi%3E%3Cmi+mathvariant%3D%22normal%22%3E%EF%BC%9A%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Et%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的“关联点”,且
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmo%3E%E2%96%B3%3C%2Fmo%3E%3Cmi%3EO%3C%2Fmi%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmsup%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmn%3E%27%3C%2Fmn%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
是等腰三角形,求t的值;
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(3)
若在直线
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
上存在点Q,使得点Q是矩形OABC关于直线
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3El%3C%2Fmi%3E%3Cmi+mathvariant%3D%22normal%22%3E%EF%BC%9A%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的“关联点”,请直接写出b的取值范围.
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