第二课时 三角形高线、中线、角平分线的计算

第二课时三角形高线、中线、角平分线的计算考点一三角形的高线例1(2023·新高考Ⅰ卷)已知在△ABC中，A＋B＝3C，2sin(A－C)＝sinB.(1)求sinA；(2)设AB＝5，求AB边上的高．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升高线问题的处理策略(1)等面积法：AD·BC＝AB·AC·sin∠BAC.(2)AD＝AB·sin∠ABD＝AC·sin∠ACD.(3)a＝c·cosB＋b·cosC.训练1(2024·咸阳模拟)在△ABC中，内角A，B，C的对边分别为a，b，c，且acosB＋eq\f(\r(3),2)b＝c.(1)求A；(2)若b＝3，c＝eq\r(3)，求△ABC中BC边上高线的长．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考点二三角形的中线例2(2024·湘潭模拟)在△ABC中，内角A，B，C的对边分别为a，b，c，满足b2(sin2B－3cos2B)＝－a(a＋b)，且sinC＝sin2B.(1)求角B的大小；(2)若△ABC的面积为2eq\r(3)，求AC边上的中线长．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升中线问题的处理策略：如图①，△ABC中，AD为BC的中线，已知AB，AC及A，求中线AD长．(1)倍长中线：如图②，构造全等，再用余弦定理即可；(2)向量法：eq\o(AD,\s\up6(→))＝eq\f(1,2)(eq\o(AB,\s\up6(→))＋eq\o(AC,\s\up6(→)))，平方即可；(3)余弦定理：邻补角余弦值为相反数，即cos∠ADB＋cos∠ADC＝0.补充：若将条件“AD为BC的中线”换为eq\f(BD,CD)＝λ”，则可以考虑方法(2)或方法(3)．训练2(2024·长沙模拟)在△ABC中，bsinB＝asinA－(b＋c)sinC.(1)求角A的大小；(2)若BC边上的中线AD＝2eq\r(3)，且S△ABC＝2eq\r(3)，求△ABC的周长．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考点三三角形的角平分线例3已知△ABC中内角A，B，C的对边分别是a，b，c，A＝60°，c＝b＋1，sinB＝eq\f(\r(21),7).(1)求c的值；(2)设AD是△ABC的角平分线，求AD的长．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升角平分线问题的处理策略：在△ABC中，AD平分∠BAC.(1)角平分线定理：eq\f(AB,AC)＝eq\f(BD,CD)；(2)利用两个小三角形面积和等于大三角形面积处理．训练3(2024·晋城模拟)已知三角形ABC的内角A，B，C的对边分别为a，b，c，且acosB＋bcosA＝2ccosB.(1)求角B；(2)若A＝eq\f(π,4)，角B的角平分线交AC于点D，BD＝eq\r(2)，求CD的长．_________________________________________________________________________________________________________________________________________