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【324208】2024八年级数学下册 专题5.4 特殊平行四边形(压轴题综合测试卷)(含解析)(新版

时间:2025-01-15 21:41:45 作者: 字数:27163字
简介:


 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 专题5.4 特殊平行四边形

学校:___________姓名:___________班级:___________考号:___________

题号

总分

得分






评卷人

得分



一.选择题(本大题共10小题,每小题3分,满分30分)

1.(沙坪坝区校级一模)下列说法正确的是(  )

A.平行四边形的对角互补 B.矩形的对角线相等且互相垂直

C.有一组邻边相等的四边形是菱形 D.有一个角是90°的菱形是正方形

【思路点拨】

根据正方形的判定,平行四边形的性质,菱形的判定与性质,矩形的性质,逐一进行判断即可.

【解题过程】

解:A.平行四边形的对角互补,错误,不符合题意;应该是平行四边形的对角相等;

B.矩形的对角线相等且互相垂直,错误,不符合题意;应该是矩形的对角线相等;

C.有一组邻边相等的四边形是菱形,错误,不符合题意;应该是有一组邻边相等的平行四边形是菱形;

D.有一个角是90°的菱形是正方形,正确,符合题意.

故选:D

2.(碑林区校级月考)已知直线abc在同一平面内,且abcab的距离为5cmbc的距离为2cm,则ac的距离是(  )

A3cm B7cm C3cm7cm D.以上都不对

【思路点拨】

因为直线c的位置不明确,所以分①直线c在直线ab外,②直线c在直线ab之间两种情况讨论求解.

【解题过程】

解:如图,①直线cab外时,

ab的距离为5cmbc的距离为2cm

ac的距离为5+27cm),

直线c在直线ab之间时,

ab的距离为5cmbc的距离为2cm

ac的距离为5﹣23cm),

综上所述,ac的距离为3cm7cm

故选:C

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

3.(阿荣旗期末)如图,在矩形ABCD中,对角线ACBD交于点O,若∠COD50°,那么∠CAD的度数是(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A20° B25° C30° D40°

【思路点拨】

只要证明OAOD,根据三角形的外角的性质即可解决问题;

【解题过程】

解:∵矩形ABCD中,对角线ACBD相交于点O

DBACODOBOAOC

OAOD

∴∠CAD=∠ADO

∵∠COD50°=∠CAD+∠ADO

∴∠CAD25°

故选:B

4.(灞桥区校级模拟)如图,在四边形ABCD中,EF分别是ADBC的中点,GH分别是对角线BDAC的中点,依次连接EGFH,连接EFGHBDEH相交于P,若ABCD,∠ABD20°,∠BDC70°,则∠GEF=(  )度.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A25 B30 C45 D35

【思路点拨】

根据三角形中位线定理得到EG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ABEGABFG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CDFGCD,根据平行线的性质求出∠EGD、∠DGF,进而求出∠EGF,再根据等腰三角形的性质、三角形内角和定理计算即可.

【解题过程】

解:∵EG分别是ADBD的中点,

EG是△ADB的中位线,

EG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ABEGAB

∴∠EGD=∠ABD20°

同理可得:FG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CDFGCD

∴∠DGF180°﹣∠BDC110°

∴∠EGF=∠EGD+∠FGD130°

ABCD

EGFG

∴∠GEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 180°﹣130°)=25°

故选:A

5.(碑林区校级三模)两张菱形贺卡如图所示叠放,其中菱形ABCD的边长为6cm,∠BAD60°,菱形A'B'C'D'可以看作是由菱形ABCD沿CA方向平移2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm得到,ADC'D'于点E,则重叠部分的面积为(  )cm2

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A8 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> B9 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> C10 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> D11 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

根据题意和题目中的数据,可以计算出ACEF的长,然后即可计算出重叠部分的面积.

【解题过程】

解:连接ACBDACBD交于点O,连接EFAC于点O,交AB于点F,如图所示,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

菱形ABCD的边长为6cm,∠BAD60°

ADAB6cmACBD,∠DAO30°

∴△DAB是等边三角形,DO3cm

AO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm),

AC6 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm

CC2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm

AC4 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm

AH2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm

EH2cm

EF4cm

重叠部分的面积为: <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 8 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> cm2),

故选:A

6.(西安二模)如图,在菱形ABCD中,∠A60°,点EF分别在边ABBC上,∠EDF60°BF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BE1,则AD的长为(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> B <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 1 C2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> D2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 1

【思路点拨】

先证明△ABD是等边三角形,再根据ASA证明△ADE≌△BDF,得到AEBF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,进而可求解AB的长,即可求解.

【解题过程】

解:∵四边形ABCD是菱形,

ABADADBC

∵∠A60°

∴△ABD是等边三角形,∠ABC180°﹣∠A120°

ADBD,∠ABD=∠A=∠ADB=∠DBC60°

∵∠EDF60°

∴∠ADE=∠BDF

在△ADE和△BDF中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△ADE≌△BDFASA),

AEBF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

BE1

ADABAE+BE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故选:B

7.(孝南区期中)已知平面直角坐标系中,有两点Aa0),B0b),且满足b <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 4PAB上一动点(不与AB重合),PEx轴,PFy轴,垂足分别为EF,连接EF,则EF的最小值为(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> B3 C4 D5

【思路点拨】

连接OP,先求出a3,则b4,再由勾股定理得AB5,然后证四边形OEPF是矩形,则EFOP,当OPAB时,OP最小,EF也最小,进而由面积法求解即可.

【解题过程】

解:如图,连接OP

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

b <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 4

a﹣3≥03﹣a≥0

a3

b4

A30),B04),

OA3OB4

∵∠AOB90°

AB <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 5

PEx轴,PFy轴,

∴∠PEO=∠PFO90°

四边形OEPF是矩形,

EFOP

OPAB时,OP最小,EF也最小,

此时,OP <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

EF的最小值为 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故选:A

8.(蒙阴县模拟)如图,平行四边形ABCD中,对角线ACBD相交于点OBD2ADEFG分别是OCODAB的中点,下列结论:①BEAC;②EGEF;③△EFG≌△GBE;④EA平分∠GEF;⑤四边形BEFG是菱形.其中正确的个数是(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A2 B3 C4 D5

【思路点拨】

由平行四边形的性质可得OBBC,由等腰三角形的性质可判断①正确,由直角三角形的性质和三角形中位线定理可判断②正确,通过证四边形BGFE是平行四边形,可判断③正确,由平行线的性质和等腰三角形的性质可判断④正确,由∠BAC≠30°可判断⑤错误.

【解题过程】

解:∵四边形ABCD是平行四边形

BODO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BDADBCABCDABCD

又∵BD2AD

OBBCODDA,且点EOC中点,

BEAC

故①正确,

EF分别是OCOD的中点,

EFCDEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CD

GRt△ABE斜边AB上的中点,

GE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ABAGBG

EGEFAGBG

故②正确,

BGEFABCDEF

四边形BGFE是平行四边形,

GFBE,且BGEFGEGE

∴△BGE≌△FEGSSS

故③正确

EFCDAB

∴∠BAC=∠ACD=∠AEF

AGGE

∴∠GAE=∠AEG

∴∠AEG=∠AEF

AE平分∠GEF

故④正确,

若四边形BEFG是菱形

BEBG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AB

∴∠BAC30°

与题意不符合

故⑤错误

故选:C

9.(涧西区一模)如图,D是平行四边形ABOC内一点,CDx轴平行,ADy轴平行,AD2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CD7 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,∠ADB135°SABD8.则点D的坐标为(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> B <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> C <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> D <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

过点BBEy轴于E点,交AD的延长线于点F,先通过AAS证出△BOE≌△CAD,根据全等三角形的性质得到OEAD2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BECD7 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,根据三角形的面积即可得到结论.

【解题过程】

解:过点BBEy轴于E点,交AD的延长线于点F

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

四边形ABOC是平行四边形,

ACOBACOB

∴∠OGC=∠BOE

ADy轴,

∴∠DAC=∠OGC

∴∠BOE=∠DAC

在△BOE和△CAD中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△BOE≌△CADAAS),

OEAD2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BECD7 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∵∠ADB135°

∴∠BDF45°

BFDF

SABD8

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ADBF8

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 8

BF4 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

EF3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

D(﹣3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 6 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ),

故选:A

10.(岳麓区校级期中)如图,在正方形ABCD中,AB6,点HDC的延长线上,连接AHBC于点F,点EBF上,且AE平分∠BAH,若CHBE,则EH等于(  )

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

A <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> B <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> C <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> D <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

过点EEGAF于点G,根据角平分线的性质可得EBEG,然后证明△EFG≌△HFCAAS),可得GFCF,证明Rt△ABE≌Rt△AGEHL),可得ABAG6,设EBEGCHmGFCFn,可得BFBCCF6﹣nAFAG+GF6+n,根据勾股定理可得n <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,根据SAFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> EFAB <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AFEG,可得EF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> m,然后根据勾股定理可得m2,在Rt△EFG中,利用勾股定理即可解决问题.

【解题过程】

解:如图,过点EEGAF于点G

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

在正方形ABCD中,BCAB6,∠B=∠BCD90°

AE平分∠BAHEGAFABBC

EBEG

CHBE

EGCH

在△EFG和△HFC中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△EFG≌△HFCAAS),

GFCF

Rt△ABERt△AGE中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

Rt△ABE≌Rt△AGEHL),

ABAG6

EBEGCHmGFCFn

BFBCCF6﹣nAFAG+GF6+n

Rt△ABF中,根据勾股定理得:

AF2AB2+BF2

6+n262+6﹣n2

解得n <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

GFCHn <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

AF6+n <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

SAFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> EFAB <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AFEG

6EF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> m

EF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> m

Rt△EFG中,根据勾股定理得:

EF2EG2+GF2

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> m2m2+ <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 2

解得m2(负值舍去),

EBEGCHm2

EF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> m <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

ECEF+FC <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 4

EH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故选:B


评卷人

得分



二.填空题(本大题共5小题,每小题3分,满分15分)

11.(南关区校级月考)▱ABCD一内角的平分线与边相交并把这条边分成5cm7cm的两条线段,则▱ABCD的周长是 3438 cm

【思路点拨】

此题注意要分情况讨论:根据角平分线的定义以及平行线的性质,可以发现一个等腰三角形,进而得到平行四边形的周长.

【解题过程】

解:如图所示:

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

在平行四边形ABCD中,ABCDADBCADBC

∴∠AEB=∠CBE

又∵BE平分∠ABC

∴∠ABE=∠CBE

∴∠ABE=∠AEB

ABAE

AE5cm时,平行四边形的周长=25+12)=34cm);

AE7cm时,平行四边形的周长=27+12)=38cm);

若点ECD边上,同理可得▱ABCD的周长为34cm38cm

综上所述,▱ABCD的周长为34cm38cm

故答案为:3438

12.(雁塔区校级四模)如图,在菱形ABCD中,对角线ACBD交于点OAC6BD8,点EOA的中点,点FBC上一点,且BF3CF,点PBD上一动点,连接PEPF,则|PFPE|的最大值为  <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>  

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

根据题意找出点E关于BD的对称点E,连接PE,构造△PFE'中的三边关系解答即可.

【解题过程】

解:在菱形ABCD中,AC6BD8

AOCO3BODO4

ABBCCDDA5

BC上取一点F,使得BF3CF,取OA的中点E,点PBD上的一动点,

E点关于BD的对称点E',连接PE'

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

PEPE'

在△PFE'中,

PFPEPFPE'FE'

则当点PFE'三点共线时,PFPE取最大值,

PFPEPF﹣[E'FE'

BC的中点H,连接HO

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

BF3CFOA的中点E

FHC的中点,E'OC的中点,

FE' <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> HO

HO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BC

FE' <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> HO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BC <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故答案为: <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

13.(铁岭月考)如图,将边长为4的等边△ABC沿射线BC平移得到△DEF,点GH分别为ACDF的中点,连接GH,点PGH的中点,连接APCP.当△APC为直角三角形时,BE 48 

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

本题先根据△APC为直角三角形进行分类讨论:①当∠APC90°时,根据直角三角形斜边中线等于斜边上的一半,即可求出PG,进而求出GHBE长度就解决了.②当∠ACP90°时,根据直角三角形中,30°角所对直角边是斜边长度的一半,可以求出PG4,进而求出GHBE长度就解决了.

【解题过程】

解:①当∠APC90°时.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∵∠APC90°MAC中点.

PGAGCG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AC2

PG2,点P是线段GH的中点.

GH2PG4

即△ABC向左平移4

BE4

当∠ACP90°时.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

GHBF

∴∠PGC=∠ACB60°

∴∠GPC30

GAC中点,AC4

CG2

Rt△GCP中,∠GCP90°,∠GPC30°

GC <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> PG

PG2CG4

P是线段GH的中点.

GH8

即△ABC向右平移8

综上所述,BE48

故答案为:48

14.(习水县模拟)如图,在△ABC中,∠BAC60°,∠ABC45°AD平分∠BACBC于点DP为直线AB上一动点.连接DP,以DPDB为邻边构造平行四边形DPQB,连接CQ,若AC6.则CQ的最小值为  <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>  

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

首先在△ACB中,由于AC4,∠CAB60°,∠CBA45°,所以可以解△CAB,即可以过CCOABO,利用三勾股定理,求出AB的长度,同理,在△DAB中,过DDHABH,可以求出DH的长度,连接DQPBM,过QQGABG,可以证明△QGM≌△DHM,所以QGDH3,由此得到Q在平行于AB的直线上运动,且距离AB两个单位长度,根据垂线段最短,可以得到当COQ三点共线时,CQ长度最小.

【解题过程】

解:如图1,过CCOABO,过DDHABH

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

Rt△ACO中,∠CAB60°

∴∠ACO30°

AO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AC3

CO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

Rt△BCO中,∠CBA45°

BOCO3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

ABAO+BO <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

AD平分∠CAB

∴∠DAB <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CAB30°

Rt△DHB中,∠CBA45°

可设DHHBa

AD2DH2a

AH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

ABAH+BH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> a+a

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

a3

DH3

如图2,过QQGABG,连接DQABM

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

四边形DPQB为平行四边形,

DMQM

在△QGM与△DHM中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△QGM≌△DHMAAS),

QGDH3

Q到直线AB的距离始终为3

所以Q点在平行于AB的直线上运动,且两直线距离为3

根据垂线段最短,

COQ三点在一条直线上时,此时CQ最小,如图3

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

最小值为:CO+3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故答案为 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

15.(北仑区一模)如图,在矩形ABCD中,AB2,点EAD的中点,点F是对角线BD上一动点,∠ADB30°,连结EF,作点D关于直线EF的对称点P,直线PEBD于点Q,当△DEQ是直角三角形时,DF的长为 133 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>  

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

分两种情况画出图形,当∠DQE90°时,如图2,如图3,当∠DEQ90°时,如图4,过点FFMAD于点M,设EMa,则FMaDM <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> a,根据直角三角形的性质即可得到结论.

【解题过程】

解:∵四边形ABCD是矩形,

∴∠BAD90°

AB2,∠ADB30°

AD2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

E是边AD的中点,

DE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

如图2,当∠DQE90°时,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

EAD的中点,

PEBD,∠ADB30°

∴∠PED60°

由对称可得,EF平分∠PED

∴∠DEF=∠PEF30°

∴△DEF是等腰三角形,

DFEF

PEBD,∠ADB30°DE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

QE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∵∠PEF30°

EF1

DFEF21

如图3

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

PEBD,∠ADB30°

∴∠PED120°

由对称可得,PFDFEPEDEF平分∠PED

∴∠DEF=∠PEF120°

∴∠EFD30°

∴△DEF是等腰三角形,

PEBD

QDQF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> DF

PEBD,∠ADB30°DE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

QE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> QD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

DF2QD3

DF的长为13

当∠DEQ90°时,如图4

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

EF平分∠PED

∴∠DEF45°

过点FFMAD于点M,设EMa,则FMaDM <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> a

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> a+a <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

a <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

DF3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

综上所述,当△DEQ是直角三角形时,DF的长为133 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故答案为:133 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>


评卷人

得分



三.解答题(本大题共8小题,满分55分)

16.(华蓥市模拟)已知:如图,在平行四边形ABCD中,点M是边AD中点,CMBA的延长线相交于点E.求证:AEAB

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

由在平行四边形ABCD中,AMDM,易证得△AEM≌△DCMAAS),即可得AECDAB

【解题过程】

证明:∵四边形ABCD是平行四边形,

ABCDABCD

∴∠E=∠DCM

M是边AD中点,

AMDM

在△AEM和△DCM中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△AEM≌△DCMAAS),

AECD

AEAB

17.(荔湾区一模)如图,正方形ABCD中,点EF分别在ADCD上,且AFBEG,连接BEAF.求证:BEAF

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

利用正方形的性质即可得到∠BAE=∠D90°ABDA,∠ABG=∠DAF,判定△ABE≌△DAFASA),即可得出BEAF

【解题过程】

证明:∵四边形ABCD是正方形,

∴∠BAE=∠D90°ABDA

∴∠DAF+∠BAF90°

又∵AFBE

∴∠ABG+∠BAF90°

∴∠ABG=∠DAF

在△ABE和△DAF中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△ABE≌△DAFASA),

BEAF

18.(亭湖区校级月考)已知:如图,在▱ABCD中,ACBD相交于点OEF分别是AOCO的中点,顺次连接BEDF

1)求证:四边形BEDF是平行四边形;

2)当ACBD满足什么关系时,四边形BEDF是矩形?为什么?

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)由平行四边形的性质得AOCOOBOD,再证OEOF,即可得出结论;

2)由平行四边形的性质结合条件证出EFBD,再由(1)得四边形BEDF是平行四边形,即可得出结论.

【解题过程】

1)证明:∵四边形ABCD是平行四边形,

AOCOOBOD

EF分别是AOCO的中点,

OE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AOOF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> CO

OEOF

四边形BEDF是平行四边形;

2)解:当AC2BD时,四边形BEDF是矩形,理由如下:

四边形ABCD是平行四边形,

AOCOOBOD

AC2BD

AOCOBD

OEOF

EFBD

由(1)得:四边形BEDF是平行四边形,

平行四边形BEDF是矩形.

19.(丹江口市模拟)如图,AMBNCBN上一点,BD平分∠ABN且过AC的中点O,交AM于点DDEBD,交BN于点E

1)求证:四边形ABCD是菱形.

2)若DEAB2,求菱形ABCD的面积.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)由ASA可证明△ADO≌△CBO,再证明四边形ABCD是平行四边形,再证明ADAB,即可得出结论;

2)由菱形的性质得出ACBD,证明四边形ACED是平行四边形,得出ACDE2ADEC,由菱形的性质得出ECCBAB2,得出EB4,由勾股定理得BD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,即可得出答案.

【解题过程】

1)证明:∵点OAC的中点,

AOCO

AMBN

∴∠DAO=∠BCO

在△AOD和△COB中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△ADO≌△CBOASA),

ADCB

又∵AMBN

四边形ABCD是平行四边形,

AMBN

∴∠ADB=∠CBD

BD平分∠ABN

∴∠ABD=∠CBD

∴∠ABD=∠ADB

ADAB

平行四边形ABCD是菱形;

2)解:由(1)得:四边形ABCD是菱形,

ACBDADCB

又∵DEBD

ACDE

AMBN

四边形ACED是平行四边形,

ACDE2ADEC

ECCB

四边形ABCD是菱形,

ECCBAB2

EB4

Rt△DEB中,由勾股定理得:BD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 2 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

S菱形ABCD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ACBD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

20.(东台市期中)如图,Rt△CEF中,∠C90°,∠CEF和∠CFE的外角平分线交于点A,过点A分别作直线CECF的垂线,点BD为垂足.

1)∠EAF 45° (直接写结果).

2)①求证:四边形ABCD是正方形.

BEEC2,求DF的长.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)根据平角的定义得到∠DFE+∠BEF360°﹣90°270°,根据角平分线的定义得到∠AFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> DFE,∠AEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BEF,求得∠AEF+∠AFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> (∠DFE+∠BEF),根据三角形的内角和定理即可得到结论;

2)①作AGEFG,如图1所示:则∠AGE=∠AGF90°,先证明四边形ABCD是矩形,再由角平分线的性质得出ABAD,即可得出四边形ABCD是正方形;

DFx,根据已知条件得到BC6,由①得四边形ABCD是正方形,求得BCCD4,根据全等三角形的性质得到BEEG2,同理,GFDFx,根据勾股定理列方程即可得到结论.

【解题过程】

1)解:∵∠C90°

∴∠CFE+∠CEF90°

∴∠DFE+∠BEF360°﹣90°270°

AF平分∠DFEAE平分∠BEF

∴∠AFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> DFE,∠AEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BEF

∴∠AEF+∠AFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> (∠DFE+∠BEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 270°135°

∴∠EAF180°﹣∠AEF﹣∠AFE45°

故答案为:45°

2)①证明:作AGEFG,如图1所示:

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

则∠AGE=∠AGF90°

ABCEADCF

∴∠B=∠D90°=∠C

四边形ABCD是矩形,

∵∠CEF,∠CFE外角平分线交于点A

ABAGADAG

ABAD

四边形ABCD是正方形;

解:设DFx

BEEC2

BC4

由①得四边形ABCD是正方形,

BCCD4

Rt△ABERt△AGE中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

Rt△ABE≌Rt△AGEHL),

BEEG2

同理,GFDFx

Rt△CEF中,EC2+FC2EF2

22+4﹣x2=(x+22

解得:x <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

DF的长为 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

21.(新郑市期末)如图,在△ABC中,点DE分别是边BCAC的中点,过点AAFBCDE的延长线于F点,连接ADCF,过点DDGCF于点G

1)求证:四边形ADCF是平行四边形;

2)若AB3BC5

AC 3 时,四边形ADCF是矩形;

若四边形ADCF是菱形,则DG  <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>  

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)由三角形中位线定理得DEAB,再证四边形ABDF是平行四边形,得AFBD,则AFDC,即可得出结论;

2)①由(1)可知,四边形ADCF是平行四边形,再由等腰三角形的性质得ADBC,则∠ADC90°,即可得出结论;

由菱形的性质得ACDFADCDBDCF,再证△ABC是直角三角形,∠BAC90°,则AC4,然后由平行四边形的性质得DFAB3,最后由菱形的面积求出DG的长即可.

【解题过程】

1)证明:∵点DE分别是边BCAC的中点,

DE是△ABC的中位线,BDCD

DEAB

AFBC

四边形ABDF是平行四边形,

AFBD

AFDC

AFBC

四边形ADCF是平行四边形;

2)解:①当AC3时,四边形ADCF是矩形,理由如下:

由(1)可知,四边形ADCF是平行四边形,

AB3AC3

ABAC

DBC的中点,

ADBC

∴∠ADC90°

平行四边形ADCF是矩形;

②∵四边形ADCF是菱形,

ACDFADCDBDCF

CFAD <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BC <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△ABC是直角三角形,∠BAC90°

AC <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 4

由(1)可知,四边形ABDF是平行四边形,

DFAB3

DGCF

S菱形ADCF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ACDFCFDG

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 4×3 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> DG

DG <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

故答案为: <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

22.(峡江县期末)如图,ABCD,点EF分别在ABCD上,连接EF,∠AEF、∠CFE的平分线交于点G,∠BEF、∠DFE的平分线交于点H

1)求证:四边形EGFH是矩形;

2)小明在完成(1)的证明后继续进行了探索,过GMNEF,分别交ABCD于点MN,过HPQEF,分别交ABCD于点PQ,得到四边形MNQP,此时,他猜想四边形MNQP是菱形,他的猜想是否正确,请予以说明.

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)根据角平分线的性质进行导角,可求得四边形EGFH的四个内角均为90°,进而可说明其为矩形.

2)根据题目条件可得四边形MNQP为平行四边形,要证菱形只需邻边相等,连接GH,由于MNEFGH,要证MNMP,只需证GHMP,只需证四边形MFHP为平行四边形,可证GH点分别为MNPQ中点,即可得出结果.

【解题过程】

1)证明:∵EH平分∠BEFFH平分∠DFE

∴∠FEH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> ,∠EFH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> DFE

ABCD

∴∠BEF+∠DFE180°

∴∠FEH+∠EFH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> (∠BEF+∠DFE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 180°90°

∵∠FEH+∠EFH+∠EHF180°

∴∠EHF180°﹣(∠FEH+∠EFH)=180°﹣90°90°

同理可得:∠EGF90°

EG平分∠AEF

EH平分∠BEF

∴∠GEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AEF,∠FEH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> BEF

AEB在同一条直线上,

∴∠AEB180°,即∠AEF+∠BEF180°

∴∠FEG+∠FEH <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> (∠AEF+∠BEF <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> 180°90°

即∠GEH90°

四边形EGFH是矩形

2)解:他的猜想正确,

理由是:

MNEFPQMPNQ

四边形MNQP为平行四边形.

如图,延长EHCD于点O

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∵∠PEO=∠FEO,∠PEO=∠FOE

∴∠FOE=∠FEO

EFFD

FHEO

HEHO

∵∠EHP=∠OHQ,∠EPH=∠OQH

∴△EHP≌△OHQ

HPHQ

同理可得GMGN

MNPQ

MGHP

四边形MGHP为平行四边形,

GHMP

MNEFMENF

四边形MEFN为平行四边形,

MNEF

GHEF

MNMP

平行四边形MNQP为菱形.

23.(海淀区校级开学)在矩形ABCD中,点P是射线BC上一动点,点B关于直线AP的对称点为E,直线PE与直线CD交于点F

1)如图,当ACE共线时,若∠ACB30°,判断△ACF的形状,并证明;

2)若当点P在线段BC上的某个位置时(不与BC重合),有∠PAF45°,求证:当点PBC延长线上任意位置时,都有∠PAF45°

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

【思路点拨】

1)根据矩形的性质和含30°角的直角三角形的性质得出AB,进而利用等边三角形的判定解答即可;

2)根据全等三角形的判定和性质和正方形的性质解答即可.

【解题过程】

1)解:△ACF为等边三角形,理由如下:

四边形ABCD为矩形,∠ACB30°

∴∠ACD60°

即∠ACF60°

Rt△ABC中,∠ACB30°

AB <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AC

B关于直线AP的对称点为E

ABAE,∠AEP90°

AE <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a> AC

即点EAC中点,

FP为线段AC的垂直平分线,

AFCF

∴△ACF为等边三角形;

2)解:当点P在线段BC上时,如图,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∵∠PAF45°

∴∠EAF45°﹣∠PAE

∵∠BAD90°

∴∠FAD45°﹣∠PAB

∴∠EAF=∠FAD

B关于直线AP的对称点为E

∴∠ABP=∠AEP90°ABAE

在△EAF和△DAF中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

∴△EAF≌△DAFAAS),

ADAE

ABAD

即矩形ABCD为正方形,

当点P在线段BC延长线上时,如图,

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矩形ABCD为正方形,

ADAE,∠ADF=∠AEF90°

Rt△ADFRt△AEF中,

 <a href="/tags/4/" title="测试" class="c1" target="_blank">测试</a> <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/58/" title="平行" class="c1" target="_blank">平行</a> <a href="/tags/82/" title="平行四边形" class="c1" target="_blank">平行四边形</a> <a href="/tags/129/" title="四边形" class="c1" target="_blank">四边形</a>

Rt△ADF≌Rt△AEFHL),

∴∠DAF=∠EAF

设∠DAFα,∠PADβ,则∠EAFα

∴∠PAE2α+β

∵∠PAE=∠PAB

∴∠PAB2α+β

∵∠PAB+∠PAD90°

2α+β+β90°

α+β45°

∴∠PAFα+β45°


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