内容正文:
第五章 相交线与平行线
5.1.2 垂线
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学习目标
【学习重点】
【学习难点】
1.理解垂线、垂线段等概念,能用三角尺或量角器过一点画已知直线的垂线;
2.理解点到直线的距离的意义,能度量点到直线的距离;
3.掌握垂线的两个性质.
掌握垂线的两条性质及其应用.
垂线的概念、性质及作图.
【学习目标】
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∠1 = 150°
1
α=30°
2
3
∠3 = 150°
∠2 = 30°
取两根木条a,b,将它们钉在一起,固定木条a,转动木条b.
b
当a与b所成角 α 为 30°时,其余的角分别为多少?
1
b
a
探究一
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∠1 = 135°
∠3 = 135°
∠2 = 45°
当a与b所成角α为45°时,其余的角分别为多少?
2
1
a
α=45°
2
3
b
b
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∠1=90°
1
α=90°
2
3
∠3=90°
∠2=90°
a
b
当a与b所成角α为90°时,其余的角分别为多少?
3
b
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两条直线相交所成的四个角中,有一个角是直角时,叫做这两条直线互相垂直.两条直线互相垂直,其中的一条直线叫做另一条直线的垂线,它们的交点叫做垂足.
AB ⊥ CD ,垂足为 O .
或 AB ⊥ CD 于点 O .
符号语言
∵ ∠AOC = 90°,
∴ AB ⊥ CD .
∴ ∠AOC = 90°.
∵ AB ⊥ CD ,
A
B
C
D
O
A
B
C
D
O
垂直的概念
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垂直是相交的一种特殊情形.
“互相垂直”是指两条直线的位置关系;
“垂线”是指其中一条直线对另一条直线的命名.
1
两条直线垂直和相交是什么关系?
2
“互相垂直”与“垂线”之间有什么区别?
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如图,直线 AB ,CD 相交于点 O ,OE ⊥ AB ,∠AOD = 125°,那么 ∠COE =_______.
A
C
E
B
D
O
125°
?
解:∵ ∠AOD = ∠BOC ,
∴ ∠BOC = ∠AOD = 125°.
∵ OE ⊥ AB ,
∴ ∠BOE = 90°,
∴ ∠COE = ∠BOC - ∠BOE
= 125°- 90°
= 35°.
35°
练习1
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l
O
如图,已知直线l,作l的垂线.
工具:直尺、三角板
A
这样画l的垂线可以画几条?
垂线的画法
无数条
探究二
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l
A
C
画:沿着三角板的另一直角边画出垂线
移:移动三角板到已知点
靠:靠三角板, 三角板的一直角边靠在直尺上
放:放直尺,直尺的一边要与已知直线重合
如图,经过直线l上一点A,画l的垂线.
这样画l的垂线可以画几条?
1条
则所画直线AC是过点A 的直线l的垂线
B
P
A
过点 P 画出射线 AB 或线段 AB 的垂线.
P
A
B
练习2
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在灌溉时,要把河中的水引到农田P处,如何挖掘能使渠道最短?
你能把这个问题转化为数学问题吗?画图试一试.
P
探究三
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假设把渠道看成线段,江河看成直线 l ,线段的一个端点是 P ,另一个端点在直线 l 上,那么原问题就转化为怎么连线的数学问题.
P
两点之间,
线段最短.
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如图PO⊥l,我们称PO为点P到直线l的垂线段.
哪一条线段最短呢?你能用一句话总结出来你观察得出的结论吗?
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垂线性质2:连接直线外一点与直线上各点的所有线段中,
垂线段最短.
简单说成:垂线段最短.
直线外一点到这条直线的垂线段的长度,叫点到直线的距离.
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其正中确的有( )
A.1个 B.2个 C.3个 D.4个
A
B
C
D
1
如图,∠BAC = 90°,AD⊥BC,垂足为D,则下列结论:
(1)AB与AC互相垂直;
(2)AD与AC互相垂直;
(3)点C到AB的垂线段是线段AB;
(4)点A到BC的距离是线段AD;
(5)线段AB的长度是点B到AC的距离;
(6)线段BD是点B到AC的垂线段.
练习3
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A.两点确定一条直线
B.两点之间,线段最短
C.过一点能作一条直线
D.在同一平面内,过一点有且只有一条直线与已知直线垂直
N
M
a
O
如图,如果直线ON⊥直线a,直线OM⊥直线a,那么OM与ON重合,其理由是( )
D
练习4
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如图,体育课上老师要测量学生的跳远成绩,其测量时主要依据是 .
A
B
l
垂线段最短
下列选项中,过点P画直线AB的垂线CD,三角尺放法正确的是( )
C
练习5
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(1)点A到BC的距离是________,
(2)点C到AB的距离是________,
(3)点B到CD的距离__________.
6
4.8
6.4
A
B
C
D
如图所示, AC⊥BC, C为垂足, CD⊥AB, D为垂足,BC = 8, CD = 4.8, BD = 6.4, AD = 3.6, AC = 6, 那么
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下列条件中,两个角的平分线互相垂直的是( )
A.互为对顶角的两个角的平分线
B.互为补角的两个角的角平分线
C.互为邻补角的两个角的角平分线
D.相邻两个角的角平分线
C
练习7
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点到直线的距离是指( )
A.从直线外一点到这条直线的垂线
B.从直线外一点到这条直线的垂线段
C.从直线外一点到这条直线的垂线的长
D.从直线外一点到这条直线的垂线段的长
D
练习8
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练习9
过线段外一点画这条线段的垂线,垂足在( )
A.这条线段上(不含端点)
B.这条线段的端点上
C.这条线段的延长线上
D.以上都有可能
D
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课堂小结
什么是垂直?
垂直和相交有什么关系?
垂线有哪些性质?
今天我们学习了哪些知识?
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课堂小结
一般情况
两条直线相交
特殊情况
对顶角相等
邻补角互补
垂线
垂线的存在唯一性
垂线段最短
点到直线的距离
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1.如图①,OC是∠AOE内的一条射线,OB是∠AOC的平分线,OD是∠COE的平分线,∠AOE=120°,求∠BOD的度数;
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2.如图②,点A,O,E在一条直线上,OB是∠AOC的平分线,OD是∠COE的平分线,请说明OB⊥OD.
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3.如图,直线AB,CD相交于点O,作∠DOE=∠BOD,OF平分∠AOE.
(1)判断OF与OD的位置关系;
(2)若∠AOC∶∠AOD=1∶5,求∠EOF的度数.
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