(2014?宝安区二模)如图,已知:在矩形ABCD中,AB=2,BC=4,P是对角线BD上的...
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发布时间：2024-10-22 11:37
共1个回答
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时间：2024-11-09 16:44
（1）如图1，
∵四边形ABCD是矩形，
∴AB=CD=2，BC=AD=4，∠C=90°
∴tan∠CBD＝CDBC＝12，
又∵∠PBF=∠CBD∠BPF=90°
∴tan∠PBF＝PFBP＝tan∠CBD＝12；
（2）∵∠EPF=∠FBP 即∠EPF=∠EBP
又∵∠PEF=∠BEP
∴△PEF∽△BEP
∴PEBE＝EFEP＝PFBP＝12
∴EP=2EF，BE=2PE 即 BE=4EF，
如图1，作EH⊥BD于H，则∠FPB=∠EHB=90°
又∵∠PBF=∠HBE
∴△PBF∽△HBE
∴HEPF＝BEBF＝43，
设BP=x，则PF＝12x
∴HE＝43PF＝43×12x＝23x
又∵CD=2，BC=4∠C=90°由勾股定理得BD＝25，
∴y＝12PD?EH＝12(25?x)?23x＝?13x2+253x
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