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7.4 平移
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1.通过具体实例认识平移，探索它的基本性质：一个图形和它经过平移所得的图形中，两组对应点的连线平行(或在同一条直线上)且相等．2．认识并欣赏平移在自然界和现实生活中的应用．3．运用图形的平移进行图案设计．
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知识点一 平移的定义
1．一般地，在平面内，将一个图形按某一方向移动一定的距离，这样的图形运动叫作平移．图形平移的方向不限于水平或竖直方向，图形可以沿平面内任何方向平移．
练习1 下列运动属于平移的是( C ).
A．转动的电风扇叶片
B．行驶的自行车后轮
C．正在上升的电梯
D．荡秋千的小朋友
知识点二 平移的性质
2．(1)新图形与原图形的形状和大小完全相同．
(2)新图形中的每一点，都是由原图形中的某一点移动后得到的，这两个点是对应点．连接各组对应点的线段平行(或在同一条直线上)且相等．
练习2 如图，三角形ABC沿射线BC方向平移到三角形DEF(点E在线段BC上).若BF＝20 cm，EC＝8 cm，则平移的距离为( A ).
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A．6 cm B．8 cm
C．12 cm D．20 cm
知识点三 利用平移进行作图
3．平移的作图步骤：
(1)确定平移条件，即平移的方向和平移的距离．
(2)找出图形中的关键点按照平移条件进行平移，得到平移前后的对应点．
(3)将平移后的对应点按照原图形进行连接．
练习3 如图，在边长为1的小正方形方格纸中，三角形ABC的顶点都在方格纸格点上．将三角形ABC平移，使点A平移到点A′处，点B′，C′分别是点B，C的对应点．
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(1)请在图中画出平移后的三角形A′B′C′；
(2)求三角形A′B′C′的面积．
【解】(1)图略
(2)三角形A′B′C′的面积为2×4－×4×1－×2×1－×2×2＝3.
知识点四 利用平移求图形的周长或面积
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练习4 如图，将直角三角形ABC沿着点B到点C的方向平移到三角形DEF的位置，平移的距离为8，AB＝14，DO＝6，则图中阴影部分的面积为( C ).
A．70 B．80
C．88 D．96
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1．下列生活现象中，属于平移现象的是( D ).
A．风车的转动
B．钟摆的摆动
C．投影片的文字经投影转换到屏幕上
D．急刹车时汽车在地面滑行
2．如图，一块边长为8米的正方形土地，在上面修了三条道路，宽都是1米，其余部分种上各种花草，则种植花草的面积是( B ).
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A．36平方米 B．42平方米
C．56平方米 D．都不对
3．如图，在三角形ABC中，∠BAC＝90°，AB＝3，AC＝4，BC＝5，将三角形ABC沿直线BC向右平移2个单位长度得到三角形DEF，连接AD.
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则下列结论：
①AC∥DF，AC＝DF；
②ED⊥AC；
③四边形ABFD的周长是16；
④AD∶EC＝2∶3.
其中正确结论的个数为( D ).
A．1 B．2
C．3 D．4
4．某酒店准备在一个楼梯铺设一种地毯，已知楼梯的宽为2米，地毯的批发价为每平方米30元，楼梯的侧面如图所示，则买地毯至少需要花费________元．
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【答案】540
5．如图，将三角形ABC沿射线AC向右平移后得到三角形CDE.如果∠BAC＝42°，∠BCA＝60°，那么∠BCD的度数是________．
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【答案】78°
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6．如图，在长方形ABCD中，AB＝7，第一次平移长方形ABCD沿AB的方向向右平移6个单位长度，得到长方形A1B1C1D1，第2次平移长方形A1B1C1D1沿A1B1的方向向右平移6个单位长度，得到长方形A2B2C2D2，…，第n次平移长方形An－1Bn－1Cn－1Dn－1沿An－1Bn－1的方向向右平移6个单位长度，得到长方形AnBnCnDn(n＞2)，若ABn的长度为2 029，则n的值为________．
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【答案】337
7．如图，网格中每个小正方形边长为1，三角形ABC的顶点都在格点上．将三角形ABC向左平移2格，再向上平移3格，得到三角形A′B′C′.(注：格点指网格线的交点)
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(1)请在图中画出平移后的三角形A′B′C′；
(2)标出平移后的三角形A′B′C′的边A′C′的中点D′；
(3)连接BB′，CC′，则这两条线段的关系是______________________；
(4)三角形ABC在整个平移过程中线段AB 扫过的面积为________；
(5)若三角形ABC与三角形ABE的面积相等，则图中满足条件且异于点C的格点E共有________个．
【答案】(1)图略
(2)图略
(3)BB′∥CC′，BB′＝CC′
(4)12
(5)9
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8．【问题情境】在综合实践课上，老师组织七年级(2)班的同学开展了探究两角之间数量关系的数学活动．如图1，已知直角三角形ABC中，∠BCA＝90°，∠BAC＝30°，直线a∥b.
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【探索发现】“快乐小组”经过探索后发现：
(1)在图1中，当∠1＝46°，求∠2的度数；
(2)不断改变∠1的度数，∠2与∠1却始终存在某种数量关系：当∠1＝50°，则∠2＝________°；当∠1＝x°时，则∠2＝________°；(用含x的代数式表示)
【实践探究】
(3)如图2，创新小组的同学将直线a向上平移，并改变∠2的位置，发现∠2与∠1也始终存在某种新的数量关系，请写出这个数量关系并说明理由．
【解】(1)如图，
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因为a∥b，所以∠2＝∠3.
因为∠1＋∠ACB＋∠3＝180°，而∠ACB＝90°，
所以∠1＋∠2＝180°－90°＝90°.
因为∠1＝46°，所以∠2＝90°－46°＝44°.
(2)40 (90－x)
(3)∠2－∠1＝120°.理由如下：
如图，过点B作BD∥a，
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则a∥b∥BD，
所以∠2＋∠ABD＝180°，∠1＝∠CBD，
因为∠ABC＝∠ABD＋∠CBD＝90°－30°＝60°，
所以60°－∠1＋∠2＝180°，
即∠2－∠1＝120°.7.4 平移
1.通过具体实例认识平移，探索它的基本性质：一个图形和它经过平移所得的图形中，两组对应点的连线平行(或在同一条直线上)且相等．2．认识并欣赏平移在自然界和现实生活中的应用．3．运用图形的平移进行图案设计．
eq \o(\s\up7(),\s\do5( ))
知识点一 平移的定义
1．一般地，在平面内，将一个图形按某一方向移动一定的距离，这样的图形运动叫作 ．图形平移的方向不限于水平或竖直方向，图形可以沿平面内任何方向平移．
练习1 下列运动属于平移的是( ).
A．转动的电风扇叶片
B．行驶的自行车后轮
C．正在上升的电梯
D．荡秋千的小朋友
知识点二 平移的性质
2．(1)新图形与原图形的形状和大小 ．
(2)新图形中的每一点，都是由原图形中的某一点移动后得到的，这两个点是 ．连接各组 的线段 (或在同一条直线上)且 ．
练习2 如图，三角形ABC沿射线BC方向平移到三角形DEF(点E在线段BC上).若BF＝20 cm，EC＝8 cm，则平移的距离为( ).
A．6 cm B．8 cm
C．12 cm D．20 cm
知识点三 利用平移进行作图
3．平移的作图步骤：
(1)确定平移条件，即平移的方向和平移的距离．
(2)找出图形中的关键点按照平移条件进行平移，得到平移前后的 ．
(3)将平移后的对应点按照原图形进行连接．
练习3 如图，在边长为1的小正方形方格纸中，三角形ABC的顶点都在方格纸格点上．将三角形ABC平移，使点A平移到点A′处，点B′，C′分别是点B，C的对应点．
(1)请在图中画出平移后的三角形A′B′C′；
(2)求三角形A′B′C′的面积．
A．70 B．80
C．88 D．96
eq \o(\s\up7(),\s\do5( ))
基础巩固
1．下列生活现象中，属于平移现象的是( ).
A．风车的转动
B．钟摆的摆动
C．投影片的文字经投影转换到屏幕上
D．急刹车时汽车在地面滑行
2．如图，一块边长为8米的正方形土地，在上面修了三条道路，宽都是1米，其余部分种上各种花草，则种植花草的面积是( ).
A．36平方米 B．42平方米
C．56平方米 D．都不对
3．如图，在三角形ABC中，∠BAC＝90°，AB＝3，AC＝4，BC＝5，将三角形ABC沿直线BC向右平移2个单位长度得到三角形DEF，连接AD.
则下列结论：
①AC∥DF，AC＝DF；
②ED⊥AC；
③四边形ABFD的周长是16；
④AD∶EC＝2∶3.
其中正确结论的个数为( ).
A．1 B．2
C．3 D．4
4．某酒店准备在一个楼梯铺设一种地毯，已知楼梯的宽为2米，地毯的批发价为每平方米30元，楼梯的侧面如图所示，则买地毯至少需要花费________元．
5．如图，将三角形ABC沿射线AC向右平移后得到三角形CDE.如果∠BAC＝42°，∠BCA＝60°，那么∠BCD的度数是________．
6．如图，在长方形ABCD中，AB＝7，第一次平移长方形ABCD沿AB的方向向右平移6个单位长度，得到长方形A1B1C1D1，第2次平移长方形A1B1C1D1沿A1B1的方向向右平移6个单位长度，得到长方形A2B2C2D2，…，第n次平移长方形An－1Bn－1Cn－1Dn－1沿An－1Bn－1的方向向右平移6个单位长度，得到长方形AnBnCnDn(n＞2)，若ABn的长度为2 029，则n的值为________．
7．如图，网格中每个小正方形边长为1，三角形ABC的顶点都在格点上．将三角形ABC向左平移2格，再向上平移3格，得到三角形A′B′C′.(注：格点指网格线的交点)
(1)请在图中画出平移后的三角形A′B′C′；
(2)标出平移后的三角形A′B′C′的边A′C′的中点D′；
(3)连接BB′，CC′，则这两条线段的关系是______________________；
(4)三角形ABC在整个平移过程中线段AB 扫过的面积为________；
(5)若三角形ABC与三角形ABE的面积相等，则图中满足条件且异于点C的格点E共有________个．
8．【问题情境】在综合实践课上，老师组织七年级(2)班的同学开展了探究两角之间数量关系的数学活动．如图1，已知直角三角形ABC中，∠BCA＝90°，∠BAC＝30°，直线a∥b.
eq \o(\s\up7(),\s\do5(图1)) eq \o(\s\up7(),\s\do5(图2))
【探索发现】“快乐小组”经过探索后发现：
(1)在图1中，当∠1＝46°，求∠2的度数；
(2)不断改变∠1的度数，∠2与∠1却始终存在某种数量关系：当∠1＝50°，则∠2＝________°；当∠1＝x°时，则∠2＝________°；(用含x的代数式表示)
【实践探究】
(3)如图2，创新小组的同学将直线a向上平移，并改变∠2的位置，发现∠2与∠1也始终存在某种新的数量关系，请写出这个数量关系并说明理由．
