【答案】
分析:設(shè)出三角形的三邊分別為a,b,c,根據(jù)正弦定理把已知的等式化簡,然后由G為三角形的重心,根據(jù)中線的性質(zhì)及向量的加法法則分別表示出
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/1.png)
和
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/2.png)
,代入化簡后的式子中,然后又根據(jù)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/3.png)
等于
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/4.png)
加
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/5.png)
,把上式進行化簡,最后得到關(guān)于
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/6.png)
和
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/7.png)
的關(guān)系式,由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/8.png)
和
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/9.png)
為非零向量,得到兩向量前的系數(shù)等于0,列出關(guān)于a,b及c的方程組,不妨令c=56,即可求出a與b的值,然后根據(jù)余弦定理表示出cosB,把a,b,c的值代入即可求出cosB的值,由B的范圍,利用特殊角的三角函數(shù)值即可得到B的度數(shù).
解答:解:因為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/10.png)
設(shè)三角形的邊長順次為a,b,c,根據(jù)正弦定理得:
56a
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/11.png)
+40b
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/12.png)
+35
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/14.png)
,
由點G為三角形的重心,根據(jù)中線的性質(zhì)及向量加法法則得:
3
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/15.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/16.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/17.png)
,3
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/18.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/19.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/20.png)
,3
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/22.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/23.png)
,
代入上式得:56a(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/24.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/25.png)
)+40b(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/26.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/27.png)
)+35c(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/28.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/29.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/30.png)
,
又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/31.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/32.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/33.png)
,上式可化為:
56a(2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/34.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/35.png)
)+40b(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/36.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/37.png)
)+35c(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/38.png)
+2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/39.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/40.png)
,
即(112a-40b-35c)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/41.png)
+(-56a-40b+70c)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/42.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/43.png)
,
則有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/44.png)
,
①-②得:168a=105c,即a:c=35:56,
設(shè)a=35k,c=56k,代入①得到b=49k,
所以cosB=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/45.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/46.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182335200928774/SYS201310241823352009287010_DA/47.png)
,又B∈(0,180°),
則B=60°.
故選D
點評:此題考查學(xué)生靈活運用正弦、余弦定理化簡求值,掌握向量的加法法則及中線的性質(zhì),是一道中檔題.