第二課時(shí) 三角形高線、中線、角平分線的計(jì)算

第二課時(shí)三角形高線、中線、角平分線的計(jì)算考點(diǎn)一三角形的高線例1(2023·新高考Ⅰ卷)已知在△ABC中，A＋B＝3C，2sin(A－C)＝sinB.(1)求sinA；(2)設(shè)AB＝5，求AB邊上的高．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升高線問(wèn)題的處理策略(1)等面積法：AD·BC＝AB·AC·sin∠BAC.(2)AD＝AB·sin∠ABD＝AC·sin∠ACD.(3)a＝c·cosB＋b·cosC.訓(xùn)練1(2024·咸陽(yáng)模擬)在△ABC中，內(nèi)角A，B，C的對(duì)邊分別為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邊上高線的長(zhǎng)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考點(diǎn)二三角形的中線例2(2024·湘潭模擬)在△ABC中，內(nèi)角A，B，C的對(duì)邊分別為a，b，c，滿足b2(sin2B－3cos2B)＝－a(a＋b)，且sinC＝sin2B.(1)求角B的大?。?2)若△ABC的面積為2eq\r(3)，求AC邊上的中線長(zhǎng)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升中線問(wèn)題的處理策略：如圖①，△ABC中，AD為BC的中線，已知AB，AC及A，求中線AD長(zhǎng)．(1)倍長(zhǎng)中線：如圖②，構(gòu)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：鄰補(bǔ)角余弦值為相反數(shù)，即cos∠ADB＋cos∠ADC＝0.補(bǔ)充：若將條件“AD為BC的中線”換為eq\f(BD,CD)＝λ”，則可以考慮方法(2)或方法(3)．訓(xùn)練2(2024·長(zhǎng)沙模擬)在△ABC中，bsinB＝asinA－(b＋c)sinC.(1)求角A的大??；(2)若BC邊上的中線AD＝2eq\r(3)，且S△ABC＝2eq\r(3)，求△ABC的周長(zhǎng)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考點(diǎn)三三角形的角平分線例3已知△ABC中內(nèi)角A，B，C的對(duì)邊分別是a，b，c，A＝60°，c＝b＋1，sinB＝eq\f(\r(21),7).(1)求c的值；(2)設(shè)AD是△ABC的角平分線，求AD的長(zhǎng)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升角平分線問(wèn)題的處理策略：在△ABC中，AD平分∠BAC.(1)角平分線定理：eq\f(AB,AC)＝eq\f(BD,CD)；(2)利用兩個(gè)小三角形面積和等于大三角形面積處理．訓(xùn)練3(2024·晉城模擬)已知三角形ABC的內(nèi)角A，B，C的對(duì)邊分別為a，b，c，且acosB＋bcosA＝2ccosB.(1)求角B；(2)若A＝eq\f(π,4)，角B的角平分線交AC于點(diǎn)D，BD＝eq\r(2)，求CD的長(zhǎng)．_________________________________________________________________________________________________________________________________________