已知直徑為8cm的圓中一弦將圓分成度數(shù)比是1:2的兩條弧,則此弦的長度為 cm.
【答案】
分析:連OA,OB,過O作OC⊥AB于C,根據(jù)垂徑定理得AC=BC,再由一弦將圓分成度數(shù)比是1:2的兩條弧,得到弧AB=360°×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/0.png)
=120°,
根據(jù)圓心角的度數(shù)等于它所對弧的度數(shù)得到∠AOB=120°,則∠A=(180°-120°)÷2=30°,在Rt△AOC中,OC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/1.png)
AB=2,AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/2.png)
OC=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/3.png)
,即可得到AB的長.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/images4.png)
解:如圖,連OA,OB,過O作OC⊥AB于C,則AC=BC,
OA=OB=4cm,
根據(jù)題意得,弧AB=360°×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/4.png)
=120°,
∴∠AOB=120°,
∴∠A=(180°-120°)÷2=30°,
在Rt△AOC中,OC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/5.png)
AB=2,AC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/6.png)
OC=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/7.png)
,
∴AB=2AC=4
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/8.png)
cm.
故答案為4
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103195809132913886/SYS201311031958091329138018_DA/9.png)
.
點評:本題考查了在同圓或等圓中,如果兩個圓心角以及它們對應(yīng)的兩條弧、兩條弦中有一組量相等,則另外兩組量也對應(yīng)相等.也考查了垂徑定理以及圓心角的度數(shù)等于它所對弧的度數(shù).