已知⊙O的半徑為10cm,弦AB∥CD,AB=12cm,CD=16cm,則AB和CD的距離為( )
A.2cm
B.14cm
C.2cm或14cm
D.10cm或20cm
【答案】
分析:本題要分類討論:
(1)AB,CD在圓心的同側(cè)如圖(一);
(2)AB,CD在圓心的異側(cè)如圖(二).
根據(jù)勾股定理和垂徑定理求解.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/images0.png)
解:(1)AB,CD在圓心的同側(cè)如圖(一),連接OD,OB,過O作AB的垂線交CD、AB于E,F(xiàn),
根據(jù)垂徑定理得ED=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/0.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/1.png)
×16=8cm,F(xiàn)B=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/2.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/3.png)
×12=6cm,
在Rt△OED中,OD=10cm,ED=8cm,由勾股定理得OE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/5.png)
=6(cm),
在Rt△OFB中,OB=10cm,F(xiàn)B=6cm,則OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/7.png)
=8(cm),
AB和CD的距離是OF-OE=8-6=2(cm);
(2)AB,CD在圓心的異側(cè)如圖(二),連接OD,OB,過O作AB的垂線交CD、AB于E,F(xiàn),
根據(jù)垂徑定理得ED=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/8.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/9.png)
×16=8cm,F(xiàn)B=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/10.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/11.png)
×12=6cm,
在Rt△OED中,OD=10cm,ED=8cm,由勾股定理得OE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/12.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/13.png)
=6(cm),
在Rt△OFB中,OB=10cm,F(xiàn)B=6cm,則OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/14.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201937022625238/SYS201311032019370226252003_DA/15.png)
=8(cm),
AB和CD的距離是OF+OE=6+8=14(cm),
AB和CD的距離是2cm或14cm.
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.