如图,△ABC中,∠C=90°,AC=BC,AD平分∠CAB交BC于点D,DE⊥AB,垂足为E,且AB=6cm,则△DEB的周长为(
如图,△ABC中,∠C=90°,AC=BC,AD平分∠CAB交BC于点D,DE⊥AB,垂足为E,且AB=6cm,则△DEB的周长为()A.4cmB.6cmC.8cmD.1...
如图,△ABC中,∠C=90°,AC=BC,AD平分∠CAB交BC于点D,DE⊥AB,垂足为E,且AB=6cm,则△DEB的周长为( )A.4cmB.6cmC.8cmD.10cm
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∵AD平分∠CAB交BC于点D
∴∠CAD=∠EADhttps://iknow-pic.cdn.bcebos.com/77094b36acaf2edd6d63da5f8e1001e9380193f2?x-bce-process=image/quality,q_85
∵DE⊥AB
∴∠AED=∠C=90
∵AD=AD
∴△ACD≌△AED.(AAS)
∴AC=AE,CD=DE
∵∠C=90°,AC=BC
∴∠B=45°
∴DE=BE
∵AC=BC,AB=6cm,
∴2BC2=AB2,即BC=
=
=3
,
∴BE=AB-AE=AB-AC=6-3
,
∴BC+BE=3
+6-3
=6cm,
∵△DEB的周长=DE+DB+BE=BC+BE=6(cm).
另法:证明三角形全等后,
∴AC=AE,CD=DE.
∵AC=BC,
∴BC=AE.
∴△DEB的周长=DB+DE+EB=DB+CD+EB=CB+BE=AE+BE=6cm.
故选B.
∴∠CAD=∠EADhttps://iknow-pic.cdn.bcebos.com/77094b36acaf2edd6d63da5f8e1001e9380193f2?x-bce-process=image/quality,q_85
∵DE⊥AB
∴∠AED=∠C=90
∵AD=AD
∴△ACD≌△AED.(AAS)
∴AC=AE,CD=DE
∵∠C=90°,AC=BC
∴∠B=45°
∴DE=BE
∵AC=BC,AB=6cm,
∴2BC2=AB2,即BC=
|
|
2 |
∴BE=AB-AE=AB-AC=6-3
2 |
∴BC+BE=3
2 |
2 |
∵△DEB的周长=DE+DB+BE=BC+BE=6(cm).
另法:证明三角形全等后,
∴AC=AE,CD=DE.
∵AC=BC,
∴BC=AE.
∴△DEB的周长=DB+DE+EB=DB+CD+EB=CB+BE=AE+BE=6cm.
故选B.
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