⊙O的半徑為5cm,弦AB∥CD,AB=6cm,CD=8cm,則AB和CD的距離是( )
A.7cm
B.8cm
C.7cm或1cm
D.1cm
【答案】
分析:因為AB、CD位置不明確,所以分在圓心的同一側(cè)和圓心兩側(cè)兩種情況討論.
解答:解:本題要分類討論:
(1)AB,CD在圓心的同側(cè),如圖①,連接OA、OC,過O作AB的垂線交CD,AB于E、F,
根據(jù)垂徑定理得ED=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/0.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/1.png)
×8=4cm,F(xiàn)B=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/2.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/3.png)
×6=3cm,
在Rt△OED中,OD=5cm,ED=4cm,由勾股定理得OE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/5.png)
=3cm,
在Rt△OFB中,OB=5cm,F(xiàn)B=3cm,則OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/7.png)
=4cm,
AB和CD的距離=OF-OE=4-3=1cm;
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/images8.png)
(2)AB,CD在圓心的異側(cè),如圖②,連接OA、OC,過O作AB的垂線交CD,AB于E、F,
根據(jù)垂徑定理得ED=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/8.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/9.png)
×8=4cm,F(xiàn)B=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/10.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/11.png)
×6=3cm,
在Rt△OED中,OD=5cm,ED=4cm,由勾股定理得OE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/12.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/13.png)
=3cm,
在Rt△OFB中,OB=5cm,F(xiàn)B=3cm,則OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/14.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101630909117667/SYS201311031016309091176012_DA/15.png)
=4cm,
AB和CD的距離是OF+OE=4+3=7cm.
AB和CD的距離是7cm或1cm.
故選C.
點評:本題涉及到垂徑定理及勾股定理,解題時要注意分類討論,不要漏解.