有梯形一腰中点时,常过此中点作另一腰的平行线,把梯形转化成平行四边形例:已知,如图,梯形abcd中,ad∥bc,e为cd中点,ef⊥ab于f,求证:s梯形abcd = ef·ab证明:过e作mn∥ab,交ad的延长线于m,交bc于n,则四边形abnm为平行四边形∵ef⊥ab∴s□abnm = ab·ef∵ad∥bc∴∠m =∠mnc又∵de = ce ∠1 =∠2∴△cen≌△dem∴s△cen = s△dem∴s梯形abcd = s五边形abned+s△cen = s五边形abned+s△dem= s梯形abcd = ef·ab有梯形一腰中点时,也常把一底的端点与中点连结并延长与另一底的延长线相交,把梯形转换成三角形例:已知,如图,直角梯形abcd中,ad∥bc,ab⊥ad于a,de = ec = bc,求证:∠aec = 3∠dae证明:连结be并延长交ad的延长线于n∵ad∥bc∴∠3 =∠n又∵∠1 =∠2 ed = ec∴△den≌△ceb∴be = en dn = bc∵ab⊥ad∴ae = en = be∴∠n =∠dae∴∠aeb =∠n+∠dae = 2∠dae∵de = bc bc = dn∴de = dn∴∠n =∠1∵∠1 =∠2 ∠n =∠dae∴∠2 =∠dae∴∠aeb+∠2 = 2∠dae+∠dae即∠aec = 3∠dae梯形有底的中点时,常过中点做两腰的平行线例:已知,如图,梯形abcd中,ad∥bc,adbc,e,f分别是ad,bc的中点,且ef⊥bc,求证:∠b =∠c证明:过e作em∥ab, en∥cd,交bc于m,n,则得□abme,□ncde∴ae = bm,ab∥= em,de = cn,cd = ne∵ae = de∴bm = cn又∵bf = cf∴fm = fn又∵ef⊥bc∴em = en∴∠1 =∠2∵ab∥em, cd∥en∴∠1 =∠b ∠2 =∠c∴∠b = ∠c任意四边形的对角线互相垂直时,它们的面积都等于对角线乘积的一半例:已知,如图,梯形abcd中,ad∥bc,ac与bd交于o,且ac⊥bd,ac = 4,bd = 3.4,求梯形abcd的面积.解:∵ac⊥bd∴s△abd =12ao·bds△bcd=12co·bd∴s梯形abcd = s△abd +s△bcd=12ao·bd+12co·bd=12(ao+co)·bd即s梯形abcd=12ac·bd =12×4×3.4=6.8答:梯形abcd面积为6.8.今天的分享就到这里,欢迎评论留言,谢谢!