如图,在⊿ABC中,AB=AC,BD平分∠ABC且∠A=36°. (1)求证:⊿DBC∽⊿BAC; (2)求sin18°
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如图,在⊿ABC中,AB=AC,BD平分∠ABC且∠A=36°. (1)求证:⊿DBC∽⊿BAC; (2)求sin18°的值.
①∵AB=AB ∠A=36°
∴∠ABC=∠C=72°
∵BD平分∠ABC
∴∠DBC=36°
∴△DBC∽△BAC
②易知∠A=∠ABD=36°
∴AD=BD=BC
AB:BC=BC:CD=BC:(AB-BC)
AB²-AB*BC-BC²=0
用求根公式(把AB当成未知数)
AB=(1+√5)BC/2 (舍负)
过A做AM⊥BC于M
则∠BAM=18°
sin18°=BM/AB=BC/2AB=BC/(1+√5)BC/2=(√5-1)/2
∴∠ABC=∠C=72°
∵BD平分∠ABC
∴∠DBC=36°
∴△DBC∽△BAC
②易知∠A=∠ABD=36°
∴AD=BD=BC
AB:BC=BC:CD=BC:(AB-BC)
AB²-AB*BC-BC²=0
用求根公式(把AB当成未知数)
AB=(1+√5)BC/2 (舍负)
过A做AM⊥BC于M
则∠BAM=18°
sin18°=BM/AB=BC/2AB=BC/(1+√5)BC/2=(√5-1)/2
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