△ABC中,sin(∠ABC/2)=根号3/3,AB=2,点D在线段AC上,且AD=2DC,BD=4倍根号3除以3,求BC的长;求三角形DBC的面
2个回答
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设CD=m ∠ADB=a
sin(∠ABC/2)=根号3/3 cos(∠ABC/2)=根号6/3
sin∠ABC=2sin(∠ABC/2)cos(∠ABC/2)=2根号2/3
cos∠ABC=1/3
由余弦定理 cosa=(4x^2+16/3-4)/(16√3m/3)
cos(180°-a)=-cosa=(x^2+16/3-BC^2)/(8√3m/3)
两式相加4x^2+16/3-4+2x^2+32/3-2BC^2=0
3x^2=BC^2-6 (1)
又由余弦定理AC^2=AB^2+BC^2-2AB*BCcos∠ABC
9x^2=4+BC^2-4BC*1/3
27x^2=12+3BC^2-4BC (2)
联立(1)(2) BC=15/2
三角形DBC的面积=(1/2)AB*BCsin∠ABC
=(1/2)*2*(15/2)*2√2/3
=5√2
sin(∠ABC/2)=根号3/3 cos(∠ABC/2)=根号6/3
sin∠ABC=2sin(∠ABC/2)cos(∠ABC/2)=2根号2/3
cos∠ABC=1/3
由余弦定理 cosa=(4x^2+16/3-4)/(16√3m/3)
cos(180°-a)=-cosa=(x^2+16/3-BC^2)/(8√3m/3)
两式相加4x^2+16/3-4+2x^2+32/3-2BC^2=0
3x^2=BC^2-6 (1)
又由余弦定理AC^2=AB^2+BC^2-2AB*BCcos∠ABC
9x^2=4+BC^2-4BC*1/3
27x^2=12+3BC^2-4BC (2)
联立(1)(2) BC=15/2
三角形DBC的面积=(1/2)AB*BCsin∠ABC
=(1/2)*2*(15/2)*2√2/3
=5√2
追问
答案BC得3额
追答
由余弦定理 cosa=(4m^2+16/3-4)/(16√3m/3)
cos(180°-a)=-cosa=(m^2+16/3-BC^2)/(8√3m/3)
两式相加4m^2+16/3-4+2m^2+32/3-2BC^2=0
3m^2=BC^2-6 (1)
又由余弦定理AC^2=AB^2+BC^2-2AB*BCcos∠ABC
9m^2=4+BC^2-4BC*1/3
27m^2=12+3BC^2-4BC (2)
联立(1)(2) BC=3 (对的,你对)
三角形ABC的面积=(1/2)AB*BCsin∠ABC
=(1/2)*2*3*2√2/3
=2√2
三角形DBC的面积=(1/3)三角形ABC的面积=2√2/3
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解 ∵sin(∠ABC/2)=√3/3<√2/2,∴∠ABC/2<45° ∠ABC<90°
∴cos∠ABC/2=√(2/3)=√6/3 sin∠ABC=2√2/3
cos∠ABC=1/3
设DC=m 那么AD=2m
在△ABC中 由余弦定理得
cos∠ADB=(AD²+BD²-AB²)/(2×AD×BD)
=(4m²+16/3-4)/(16m√3/3)①
在△BDC中
cos∠BDC=(DC²+BD²-BC²)/2×DC×BD
=(m²+16/3-BC²)/(8m√3/3)②
∵∠ADB+∠BDC=180°
∴cos∠ADC=-cos∠BDC cos∠ADC+cos∠BDC=0
①+②(4m²+16/3-4)/(16m√3/3)+(m²+16/3-BC²)/(8m√3/3)=0
(6m²-12-2BC²)/(16m√3/3)=0
3m²=BC²-6③
在△ABC中 由余弦定理得
AC²=AB²+BC²-2•AB•BC•cos∠ABC
9m²=4+BC²-4BC•(1/3)
27m²=12+3BC²-4BC④
③×9-④整理得
3BC²+2BC-33=0
(BC-3)(3BC+11)=0
BC=3(-11/3不合题意舍去)
∴cos∠ABC/2=√(2/3)=√6/3 sin∠ABC=2√2/3
cos∠ABC=1/3
设DC=m 那么AD=2m
在△ABC中 由余弦定理得
cos∠ADB=(AD²+BD²-AB²)/(2×AD×BD)
=(4m²+16/3-4)/(16m√3/3)①
在△BDC中
cos∠BDC=(DC²+BD²-BC²)/2×DC×BD
=(m²+16/3-BC²)/(8m√3/3)②
∵∠ADB+∠BDC=180°
∴cos∠ADC=-cos∠BDC cos∠ADC+cos∠BDC=0
①+②(4m²+16/3-4)/(16m√3/3)+(m²+16/3-BC²)/(8m√3/3)=0
(6m²-12-2BC²)/(16m√3/3)=0
3m²=BC²-6③
在△ABC中 由余弦定理得
AC²=AB²+BC²-2•AB•BC•cos∠ABC
9m²=4+BC²-4BC•(1/3)
27m²=12+3BC²-4BC④
③×9-④整理得
3BC²+2BC-33=0
(BC-3)(3BC+11)=0
BC=3(-11/3不合题意舍去)
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