相似三角形的基本模型(一线三等角)

模型中的相似三角形（2）【基本模型】1.如图，∽（一线三等角）如图，∽（一线三直角）如图，特别地，当是中点时：∽∽EMBEDEquation.3EMBEDEquation.3平分，平分。2.一线三等角辅助线添加：一般情况下，已知一条直线上有两个等角（直角）或一个直角时，可构造“一线三等角”型相似。【巩固提高】1.已知中，是的中点，边上有一点延长线上有一点，使已知，则提示：，是的中点∴由∽∴,2.如图，等边中，是边上的一点，且，把折叠，使点落在边上的点处．那么的值为　　　　．提示：由翻折可得：设：则∵∽，∴3.在矩形中，，，把矩形沿直线翻折，点落在边上的点处，若，那么的长等于提示：作于，则∵∽,∴∵∴4.在矩形中，，点在边上，联结，△沿直线翻折后点落到点，过点作，垂足为点,如果，那么．提示：作过点作∥，分别交、于、。∵，∴设，由翻折可得：∵∽∴,即∴，∴5.已知△，，，边长，点在上，且，点是上一动点，联结，将线段绕点逆时针旋转得到线段，要使点恰好落在上，则的长是。提示：构造“一线三等角”∴△≌△∴，，∴6.如图，已知∥，，，点是射线上的一个动点（点与点不重合），点是线段上的一个动点（点与点、不重合），联结，过点作的垂线，交射线于点，联结．设，．（1）当时，求关于的函数关系式，并写出它的定义域；（2）在（1）的条件下，取线段的中点，联结，若，求的长；（3）如果动点、在运动时，始终满足条件，那么请探究：的周长是否随着动点、的运动而发生变化？请说明理由．解：(1)∵∴∽∴∴（）（2）过作，垂足为∵是线段的中点，，∴,，，∴，（3）∵又∴又∽∴∴∴7.如图，已知中，,,是的中点，将角的顶点置于点，并绕点旋转，使角的两边分别交边、于点、，连接.(1)求证∽；(2)设，试用关于的式子表示。解：（1）∵,∴,∵∴∴∽∴∵∴,即,∵∴∽（2）作于，于,于∵,,∴同理：∵∽∴∵，∴∵∽∴∵于，于∴同理：∴_1460891198.unknown_1460891251.unknown_1460891252.unknown_1460891264.unknown_1460891265.unknown_1460891266.unknown_1460891375.unknown_1460891385.unknown_1460891386.unknown_1460891388.unknown_1460891394.unknown_1460891675.unknown_1460891838.unknown_1460894617.unknown_1471942342.unknown_1471942347.unknown_1471942353.unknown_1471942380.unknown_1471942388.unknown_1471942392.unknown_1471942400.unknown_1471942404.unknown_1471942410.unknown_1471942414.unknown_1471942420.unknown_1471942424.unknown_1471942429.unknown_1471942434.unknown_1471942439.unknown_1471942444.unknown_1471942449.unknown_1471942455.unknown_1471942459.unknown_1471942464.unknown_1471942473.unknown_1471942481.unknown_1471942482.unknown_1471942488.unknown_1471942497.unknown_1471942506.unknown_1471942514.unknown_1471942519.unknown_1471942524.unknown_1471942531.unknown_1471942537.unknown_1471942543.unknown_1471942548.unknown_1471942558.unknown_1471942578.unknown_1471942582.unknown_1471944443.unknown_1471944472.unknown_1471944473.unknown_1471944484.unknown_1471944485.unknown_1471944517.unknown_1471944538.unknown_1471944652.unknown_1471944685.unknown_1471944830.unknown_1471944896.unknown_1471945138.unknown_1471945139.unknown_1471945152.unknown_1471945164.unknown_1471945183.unknown_1471945203.unknown_1471945225.unknown_1471945261.unknown_1471945381.unknown_1471945402.unknown_1471945470.unknown_1471945511.unknown_1471945561.unknown_1493047686.unknown_1493047749.unknown_1493047757.unknown_1493047770.unknown_1493047804.unknown_1493047810.unknown_1493047822.unknown_1493047861.unknown_1493047871.unknown_1493047881.unknown_1493047899.unknown_1493047915.unknown_1493047930.unknown_1493047938.unknown_1493047944.unknown_1493103152.unknown_1493196007.unknown_1494852148.unknown_1494852160.unknown_1496051781.unknown_1521633833.unknown_1521633875.unknown_1534313444.unknown_1534313650.unknown_1534579729.unknown_1534579783.unknown_1534579807.unknown_1534579898.unknown_1534662814.unknown_1534662829.unknown_1534663360.unknown_1534663461.unknown_1534663555.unknown_1534663663.unknown_1534665571.unknown_1534665572.unknown_1534665573.unknown_1534665649.unknown_1534665650.unknown_1534665729.unknown_1534666036.unknown_1534666100.unknown_1534666110.unknown_1534666178.unknown_1534666222.unknown_1534666288.unknown_1534666312.unknown_1534666313.unknown_1534666314.unknown_1534666315.unknown_1534666316.unknown_1534666317.unknown_1534666318.unknown_1534666319.unknown_1534666320.unknown_1534666321.unknown_1534666341.unknown_1534666344.unknown_1534666345.unknown_1534666348.unknown_1534666351.unknown_1534666353.unknown_1534666356.unknown_1534666360.unknown_1534666362.unknown_1534666365.unknown_1534666368.unknown_1534666370.unknown_1534666373.unknown_1534666379.unknown_1534666435.unknown_1534666458.unknown_1534666477.unknown_1534666507.unknown_1534666524.unknown_1534666534.unknown_1534666553.unknown_1534666617.unknown_1534666696.unknown_1534666737.unknown_1534666766.unknown_1534666786.unknown_1534666807.unknown_1534666908.unknown_1534666916.unknown_1534666991.unknown_1534668113.unknown_1534668134.unknown_1534668198.unknown_1534668223.unknown_1534668247.unknown_1534668292.unknown_1534668381.unknown_1534674518.unknown_1534765908.unknown_1534765909.unknown_1534766086.unknown_1534766087.unknown_1534766090.unknown_1534766091.unknown_1534766103.unknown_1534766106.unknown_1534766118.unknown_1534766121.unknown_1534766158.unknown_1534766212.unknown_1534766588.unknown_1534766637.unknown_1534766658.unknown_1534766685.unknown_1534766722.unknown_1534766754.unknown_1534768865.unknown_1534768917.unknown_1534768964.unknown_1534768965.unknown_1534932057.unknown_1534932368.unknown_1534932393.unknown_1534932474.unknown_1534932486.unknown_1534932498.unknown_1534932531.unknown_1534932536.unknown_1534932562.unknown_1534932563.unknown_1534932565.unknown_1534932566.unknown_1534932574.unknown_1534932575.unknown_1534932599.unknown_1534932607.unknown_1534932635.unknown_1534932636.unknown_1534932642.unknown_1534932643.unknown_1534932859.unknown_1534933269.unknown_1534933270.unknown_1534933271.unknown_1534933309.unknown_1534933310.unknown_1534933334.unknown_1534933335.unknown_1534933465.unknown_1534933466.unknown_1534933469.un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