“图形的旋转”题型全攻略

书书书第七章!旋!转第七章!旋!转图形的旋转是近几年中考必考的内容!运用旋转的全等变换!证明线段相等和差倍分关系以及角相等和差倍分关系都是近几年中考常见的题型!第一节!旋转的性质经过旋转!对应线段相等!对应角相等#任意两条对应线段所在直线的夹角都等于旋转角#旋转前后的图形全等!!!旋转的基本图形有$%!&如图!将#$旋转至%#$%!则#%$#$%!%#&如图!将##$旋转至#%#$%!连结%!$$%则##%$#$#$%!#!利用旋转性质的解题步骤为$%!&找旋转点!得等边等角#%#&证全等或相似#%$&利用全等或相似得到边角关系!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例!!在#$&中!$%!&$&&!将#$&绕点按顺时针方向旋转!得到#’(!旋转角为!%’(&!&!)’(&!点$的对应点为点’!点&的对应点为点(!连结$’!$(!%!&如图!当!%’(时!延长$(交’于点)!求证$$)’’!)’)!!!!!中考数学压轴题破解策略%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!%#&在旋转的过程中!过点’作’*’$于点*!连结&(!当’*&$!且线段’*与线段(无公共点时!求$(*&(!解!!!由旋转的性质可得$’#(’(#$’!%’(!所以#$’为等边三角形#所以$’$#从而得到$(为’的垂直平分线#所以$)’’#)’)!!#如图#按照题意画出图形#令&(与$的交点为+!由旋转的性质可得&&$((’#&$&$(’(’!因为’**’(*($&$*&$*&$!)’(#且已知’*&$#所以&$($!所以$#&(互相垂直平分#则&&$$((!所以$(*&(&*#+&#,$槡#!$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!!例!如图!已知#$&是等边三角形!点(在线段$上!点’在直线$&上!且(’(&!将#$&(绕点&顺时针旋转%’(至#&)!连结()!%!&求证$$’$*)#%#&若点(在线段$的延长线上!其他条件不变!线段$!’$!)之间有怎样的数量关系’请说明理由#%$&若点(在线段$的延长线上!其他条件不变!线段$!’$!)之间有怎样的数量关系’请说明理由!证明!!!如图#过点(作(*(&#交$&于点*#则#($*为等边三角形!易证#($’)#(*&#所以’$&*(!由旋转的性质可得)$(#所以$$(*()*’$!!#$’$,)!理由如下$如图#过点(作(*(&#交&’于点*#则#($*为等边三角形!易证#(*’)#($&#所以’*$&$!由旋转的性质可得)$($*#所以$’*’$,$*’$,)!第七章!旋!转!!!$$),’$!理由如下$如图#过点(作(*(&#交$&延长线于点*#则#($*为等边三角形!易证#($’)#(*&#所以’$&*(!由旋转的性质可得)$(#所以$$(,(),’$!!!如图!将正五边形$&’(绕点顺时针旋转%’(后!旋转前后两图形有另一交点#!连结#再将#所在的直线绕点逆时针旋转%’(后!交旋转前的图形于点,!连结,#!判断##,的形状!并说明理由!#!如图!在菱形$&’中!&#!$’#槡$!&!$’相交于点#!将一个足够大的直角三角板%’(角的顶点放在菱形$&’的顶点处!绕点左右旋转!其中三角板%’(角的两边分别与边$&!&’相交于点(!)!连结()!与&相交于点*!#!$判断#()是哪一种特殊三角形!并说明理由##$旋转过程中!当点(为边$&的四等分点时#$(*&($!求&*的长!第二节!!!形模型当图形具有邻边相等这一特征时!可以把图形的某部分绕其邻边的公共端点旋转到另一位置!将分散的条件相对集中起来!从而解决问题!因为正方形正三角形的边长相等!所以在这两种图形中常常应用旋转变换!%!&如图!等边#$&内有一点,!连结,!$,!&,!将#$,&绕点$逆时针旋转%’(得到#$,%!则#$,,%是等边三角形##,,%的形状由,!$,!&,的长度决定!#!!!中考数学压轴题破解策略%#&如图!正方形$&’内有一点,!连结,!$,!&,!将#$,&绕点$逆时针旋转-’(得到#$,%!则#$,,%是等腰直角三角形##,,%的形状由,!$,!&,的长度决定!这类题目中不提旋转!而是通过旋转添加辅助线!从而解决问题!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例!!在#$&中!$&%’(!%!&如图!!若$&!点,在#$&内!且,$!,&.!,&!&’(!求,$的长#%#&如图#!若$&!点,在#$&外!且,$!,$&!,&.!求,&的度数#%$&如图$!若$#&!点,在#$&内!且,槡$!,$&!,&!#’(!求,&的长!解!!!如图#将#,&绕点顺时针旋转%’(#得到#-$#连结,-!易证#,-是等边三角形!从而在#,-$中#有,-$-’(#,-$#$-.#所以,$&!!#如图#将#,&绕点顺时针旋转%’(#得到#-$#连结,-!易证#,-是等边三角形!从而在#,-$中#有,-$#$-.#,$&#所以,-$-’(#从而,&-$$’(!!$如图#作#-&#使得-!#,#&-!#$,#连结,-!易证#&$$#-,!从而在#-,&中#有-,&-’(#,-$##-&&##所以,&#!$第七章!旋!转%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例!如图!正方形$&’外有一点(!满足(’(&!且’(!&(!求证$#’(&为等边三角形!证明!如图#过点’作’)’’(#且’)’(#连结&)交(于点*#连结()!易证#’()#&’)#所以’)&’(!&(#从而)*()’(-’(#*)($’(!所以&(!#()槡##’)槡##&(#所以*(&.&(#’(&%’(#即#’(&为等边三角形!!!如图!点,在等边#$&中内!,#!,$#槡$!,&.!则#$&的边长是!#!#!$如图!!在正方形$&’内有一点,!,槡&!,$槡#!,&!!则$,&的度数为##$如图#!在正六边形$&’()内有一点,!,#槡!$!,$.!,&#!则$,&的度数为!正六边形$&’()的边长为!!!!!$!如图!在#$&中!$&!’!$&.&(!$&&’!$&’-’(!求’的长!%!!!中考数学压轴题破解策略第三节!中心对称模型通常情况下!遇到线段中点时!可将图形绕该点旋转!)’(!构造中心对称!通常所说的倍长中线!实际上就是作中心对称!中心对称模型的基本图形有$%!&倍长中线!构造全等三角形!如图!.为$&中点!延长.至点’!使得’..!连结&’!则#$.)#’&.#%#&倍长类中线%与中点有关的线段&!构造全等三角形!如图!.为$&中点!延长’.至点(!使得(.’.!连结$(!则#&’.)#$(.!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例!!正方形$&’的边长为#!点,是射线&$上一个动点!连结,!,’!点.!/分别为$&!,的中点!连结./交,’于点-!点,%与点,关于直线$对称!且点,%在线段$&上!连结,%!若点-恰好在直线,%上!求$,的长!解!如图#延长./#’交于点(!则#(/)#,./#所以(,.$,*!!易证#-’$#,%-,##(-’$#.-,#所以,,%’,-’-,.’(#从而#$,#$,*!$,*$!解得$,槡#,!!&第七章!旋!转%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例!在/0#&$和/0#()中!&$()-’(!若,是$)的中点!连结,&!,(!%!&如图!!若点(!)分别落在边$!&上!请直接写出此时线段,&与,(的数量关系#%#&如图#!把图!中的#()绕着点顺时针旋转!当点(落在边&的延长线上时!上述结论是否成立’若成立!请给予证明#若不成立!请说明理由#%$&如图$!若点)落在边$上!则上述结论是否仍然成立’若成立!请给予证明#若不成立!请说明理由!解!!!易得,&,(!#$)#即,&与,(相等!!#结论成立!理由如下$如图#延长&,交()的延长线于点’#则$&()’!易证#$,&)#),’#所以,&,’!而&(’-’(#所以,(!#&’,&!!$结论仍然成立!理由如下$如图#过点)作)’($&#交&,延长线于点’!易得,’,&#)’$&!所以(&()$&())’!而)(,$&,)’#所以(&!)’(,#)(()’!如图#连结&(#(’!则#(&$#()’#所以(&)(’#&(’()-’(!所以,(!#&’,&!’!!!中考数学压轴题破解策略%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!例#!已知#$&是等腰三角形!$&-’(!&’!#$&!’(’&(!’(&(!连结(!点.是(的中点!%!&如图!!若点’在#$&的内部!连结$’!点/是$’中点!连结./!/(!求证$./’(#%#&如图#!将图!中的#&’(绕点&逆时针旋转!使$&’$’(!连结$’!点/是$’中点!连结./!探索./&的值!解!!!如图#延长(/至点)#使得/)/(#连结)$!易证#’(/)#$)/#从而可得$)(’(#$)’(!延长)$#&(交于点*#则*-’(#从而#$#*#&四点共圆#所以$)&(!连结)#所以#$))#&(!00!所以)(#且)’(!而./()#所以./!#(#且./’(!!#如图#同!!可得./!#(#且./’(!由题意可得&#&(#作(+’&于点+#则(&+%’(#所以&+!#(&!.&#(+槡$.&#从而(+#*(+槡#槡1#&#所以./&槡1.!!!已知#$&和#’(是等腰直角三角形!&$’(-’(!点)为$(中点!连结’)!&)!#!$如图!!当点’在$上!点(在&上!请直接写出此时线段’)!&)的数量关系和位置关系#不用证明$##$如图#!在#!$的条件下将#’(绕点顺时针旋转.&(时!请你判断此时#!$中的结论是否仍然成立!并证明你的判断(#第七章!旋!转#$$如图$!在#!$的条件下将#’(绕点顺时针旋转!时!请你判断此时#!$中的结论是否仍然成立!并证明你的判断!#!如图!!在菱形$&’和$()*中!点!$!(在同一条直线上!,是线段’)的中点!连结,*!,&!若$&$()%’(!#!$请写出线段,